less than or equal to - Shows an inequality of one value being equal to or less than another
Formula: ≤

Free 1 Die Roll Calculator - Calculates the probability for the following events in the roll of one fair dice (1 dice roll calculator or 1 die roll calculator):
* Probability of any total from (1-6)
* Probability of the total being less than, less than or equal to, greater than, or greater than or equal to (1-6)
* The total being even
* The total being odd
* The total being a prime number
* The total being a non-prime number
* Rolling a list of numbers i.e. (2,5,6)
* Simulate (n) Monte Carlo die simulations.
1 die calculator

Free 2 dice roll Calculator - Calculates the probability for the following events in a pair of fair dice rolls:
* Probability of any sum from (2-12)
* Probability of the sum being less than, less than or equal to, greater than, or greater than or equal to (2-12)
* The sum being even
* The sum being odd
* The sum being a prime number
* The sum being a non-prime number
* Rolling a list of numbers i.e. (2,5,6,12)
* Simulate (n) Monte Carlo two die simulations. 2 dice calculator

2 is greater than or equal to w and -7 is less than or equal to w Written as an inequality, we have: -7 <= w <= 2

2 less than 3 times n is 4 more than n 3 times n: 3n 2 less than 3 times n 3n - 2 4 more than n: n + 4 The word [I]is[/I] means equal to, so we set 3n - 2 equal to n + 4: [B]3n - 2 = n + 4[/B]

3x less than 2 times the sum of 2x and 1 is equal to the sum of 2 and 5 This is an algebraic expression. Let's take this algebraic expression in 5 parts: [LIST=1] [*]The sum of 2x and 1 means we add 1 to 2x: 2x + 1 [*]2 times the sum of 2x and 1: 2(2x + 1) [*]3x less than the sum of 2x and 1 means we subtract 3x from 2(2x + 1): 2(2x + 1) - 3x [*]The sum of 2 and 5 means we add 5 to 2: 2 + 5 [*]Finally, the phrase [I]equal[/I] means an equation, so we set #3 equal to #4 [/LIST] Our algebraic expression is: [B]2(2x + 1) - 3x = 2 + 5[/B] Now, some problems may ask you to simplify. In this case, we multiply through and group like terms: 4x + 2 - 3x = 7 [B]x + 2 = 7 <-- This is our simplified algebraic expression [/B] Now, what if the problem asks you to solve for x, [URL=' type this into our search engine[/URL] and get: x =[B] 5 [MEDIA=youtube]3hzyc2NPCGI[/MEDIA][/B]

4 times x plus 2 is at most 10 4 times x 4x Plus 2 4x + 2 At most means less than or equal to, so we have: [B]4x + 2 <= 10[/B]

-2 times a number x is written as -2x. Less means subtract, so we have 7 less than this is -2x - 7. Finally, greater than or equal to is >=, so our expression becomes: -2x - 7 >= 41

-2 times a number x is denoted as -2x. 7 less than that means we subtract 7: -2x - 7 Finally, that entire expression is greater than or equal to 41 -2x - 7 >= 41

-2 times a number x is denoted as -2x. 7 less means we subtract, so 7 less than that is -2x - 7. Finally, that entire expression is greater than or equal to 41 -2x - 7 >= 41

7 less than -2 times a number x is greater than or equal to 41 -2 times a number x -2x 7 less than this -2x - 7 Now we set this expressions greater than or equal to 41 [B]-2x - 7 >= 41[/B]

8 less than X means X - 8. The word is means equal to. So we have: X - 8 = 31

8 more than twice a number is less than 6 more than the number. This is an algebraic expression, let's take it in pieces... The phrase [I]a number[/I] means an arbitrary variable, let's call it x. 8 more than twice a number: Twice a number means multiply x by 2: 2x Then add 8: 2x + 8 6 more than the number, means we add 6 to x x + 6 The phrase [I]is less than[/I] means an inequality, where we set 2x + 8 less than x + 6 [B]2x + 8 < x + 6[/B]

9 less than 5 times a number is 3 more than 2x The phrase [I]a number[/I] means an arbitrary variable, let's call it x: x 5 times a number means we multiply x by 5: 5x 9 less than 5x means we subtract 9 from 5x: 5x - 9 3 more than 2x means we add 3 to 2x: 2x + 3 The word [I]is[/I] means an equation, so we set 5x - 9 equal to 2x + 3: [B]5x - 9 = 2x + 3 <-- This is our algebraic expression[/B] [B][/B] If you want to solve for x, [URL=' this equation into the search engine[/URL], and we get: x = [B]4[/B]

9 less than twice x is twice y Twice x means we multiply x by 2: 2x 9 less than Twice x means we subtract 9 from 2x 2x - 9 Twice y means we multiply y by 2: 2y The word [I]is[/I] means equal to, so we set 2x - 9 equal to 2y: [B]2x - 9 = 2y[/B]

A bakery has a fixed cost of $119.75 per a day plus $2.25 for each pastry. The bakery would like to keep its daily costs at or below $500 per day. Which inequality shows the maximum number of pastries, p, that can be baked each day. Set up the cost function C(p), where p is the number of pastries: C(p) = Variable Cost + Fixed Cost C(p) = 2.25p + 119.75 The problem asks for C(p) at or below $500 per day. The phrase [I]at or below[/I] means less than or equal to (<=). [B]2.25p + 119.75 <= 500[/B]

A bank charges a service fee of $7.50 per month for a checking account. A bank account has $85.00. If no money is deposited or withdrawn except the service charge, how many months until the account balance is negative? Let m be the number of months. Our balance is denoted by B(m): B(m) = 85 - 7.5m The question asks when B(m) is less than 0. So we set up an inequality: 85 - 7.5m < 0 To solve this inequality for m, [URL=' type it in our search engine[/URL] and we get: m > 11.3333 We round up to the next whole integer and get [B]m = 12[/B]

A car whos original value was $25600 decreases in value by $90 per month. How Long will it take before the cars value falls below $15000 Let m be the number of months.We have our Book Value B(m) given by: B(m) = 25600 - 90m We want to know when the Book value is less than 15,000. So we have an inequality: 25600 - 90m < 15000 Typing [URL=' inequality into our search engine and solving for m[/URL], we get: [B]m > 117.78 or m 118 months[/B]

a carnival charges $6 admission and $2.50 per ride. You have $50 to spend at the carnival. Which of the following inequalities represents the situation if r is the number of rides? We set up our inequality using less than or equal to, since our cash is capped at $50. We use S for our : Cost per ride * r + Admission <= 50 Plugging in our numbers, we get: 2.50r + 6 <= 50 [B][/B] Now, if the problem asks you to put this in terms of r, then [URL=' plug this inequality into our search engine[/URL] and we get: r <= 17.6 Since we cannot do fractional rides, we round down to 17: [B]r <= 17[/B]

A certain species of fish costs $3.19 each. You can spend at most $35. How many of this type of fish can you buy for your aquarium? Let the number of fish be f. We have the following inequality where "at most" means less than or equal to: 3.19f <= 35 [URL=' this inequality into our search engine[/URL], we get: f <= 10.917 Since we need a whole number of fish, we can buy a maximum of [B]10 fish[/B].

A daily pass costs $62. A season ski pass costs $450. The skier would have to rent skis with either pass for $30 per day. How many days would the skier have to go skiing in order to make the season pass less expensive than the daily passes? Let d be the number of days the skier attends. Calculate the daily cost: Daily Total Cost = Daily Cost + Rental Cost Daily Total Cost = 62d + 30d Daily Total Cost = 92d Calculate Season Cost: Season Total Cost = Season Fee + Rental Cost Season Total Cost = 450 + 30d Set the daily total cost and season cost equal to each other: 450 + 30d = 92d [URL=' this equation into the search engine[/URL], we get d = 7.258. We round up to the next full day of [B]8[/B]. Now check our work: Daily Total Cost for 8 days = 92(8) = 736 Season Cost for 8 days = 30(8) + 450 = 240 + 450 = 710. Therefore, the skier needs to go at least [B]8 days[/B] to make the season cost less than the daily cot.

A is the set of integers greater than or equal to -5 and less than or equal to -2 [B]-5 <= A <= -2[/B]

A number m is no less than -8 and fewer than 9. No less than means greater than or equal to: m >= -8 Fewer than 9 means less than 9: m < 9 Combine these two inequalities to get [B]-8 <= m < 9[/B]

A number n is no less than 2 and no more than 49. This is a compound inequality. Let's break it into parts. Step 1: No more than 49 means 49 or less. Or, less than or equal to 49 <= 49 Step 2: no less than 2 means 2 or greater. Or, greater than or equal to 2 >=2 Writing this in terms of the number n, we have: [B]2 <= n <= 49[/B]

A number t is no less than 30 and fewer than 40. This is a compound inequality. Take it in 3 parts: Step 1: fewer than 40 means less than (does not include 40) t < 40 Step 2: no less than 30 means greater than or equal to t >= 30 Step 3: Combine these 2 statements into one compound inequality: [B]30 <= t < 40[/B]

A pair of dice is rolled. Find the probability of rolling a sum of not less than 5. The phrase [I]not less than[/I] also means greater than or equal to. So we [URL=' our 2 dice calculator for a sum roll of 5 or greater[/URL] and we get: [B]5/6[/B]

A quantity x is at least 10 and at most 20 The phrase [I]at most[/I] means less than or equal to The phrase [I]at least[/I] means greater than or equal to. So we have the following inequality [B]10 <= x <= 20[/B]

A sweater that you love costs $32. You really want the sweater but only have $35. If theres a sales tax of 4% on the item, do you have enough to buy the sweater? Calculate after-tax amount: After tax amount = Sale Price * (1 + sales tax percent) After tax amount = 32 * (1 + 0.04) <-- Since 4% = 0.04 After tax amount = 32 * (1.04) After tax amount = $33.28 [B]Yes[/B], since $33.28 is less than or equal to $35, you have enough to buy the sweater.

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to spend less than $12 on a ride. Which inequality can be used to find the distance Samantha can travel? [LIST] [*]Each ride will cost 1.50 + 0.8x where x is the number of miles per trip. [*]This expression must be less than 12. [/LIST] [U]Setup the inequality:[/U] 1.5 + 0.8x < 12 [U]Subtracting 1.5 from each side of the inequality[/U] 0.8x < 10.5 [U]Simplifying even more by dividing each side of the inequality by 0.8, we have:[/U] [B]x < 13.125[/B]

absolute value of x is less than or equal to 4 Absolute value of x: |x| Set up an inequality where this is less than or equal to 4: [B]|x| <= 4 [/B] <-- This is our algebraic expression To solve this, we have the following compound inequality: -4 < x < 4

These are now available as shortcuts. You can type any number or variable in the following forms: [LIST] [*]Absolute value of x less than 8 [*]Absolute value of x less than or equal to 8 [*]Absolute value of x greater than 8 [*]Absolute value of x greater than or equal to 8 [*]Absolute value of x equal to 8 [/LIST]

We have two expressions here, so we need a union since we have the word [U]or[/U]. First, All real numbers less than or equal to -1 is x <= -1. All real numbers greater than 5 is x > 5 So we have x <= -1 U x > 5 [MEDIA=youtube]boOueZTCSuU[/MEDIA]

Amanda runs each lap in 4 minutes. She will run less than 44 minutes today. What are the possible number of laps she will run today? Notes for this problem: [LIST] [*]Let laps be l. [*]Lap time = Time per lap * number of laps (l) [*]Less than means we have an inequality using the < sign [/LIST] We have the inequality: 4l < 44 To solve this inequality for l, we [URL=' it in our math engine[/URL] and we get: [B]l < 11[/B]

An elevator can hold less than 2700 pounds of extra weight. If an average person weighs 150 pounds, what is the maximum number of people (p) that can be on the elevator at one time? Total weight = average weight per person * Number of people Total weight = 150p We know from the problem that: 150p < 2700 We want to solve this inequality for p. Divide each side of the inequality by 150: 150p/150 < 2700/150 Cancel the 150's on the left side and we get: p < [B]18[/B]

An elevator has a maximum weight of 3000 pounds. How many 150-pound people can safely ride the elevator? (Use "p" to represent the number of people) Maximum means less than or equal to. We have the inequality: 150p <= 3000 To solve this inequality for p, we [URL=' it in our search engine[/URL] and we get: [B]p <= 20[/B]

Angelicas financial aid stipulates that her tuition cannot exceed $1000. If her local community college charges a $35 registration fee plus $375 per course, what is the greatest number of courses for which Angelica can register? We set up the Tuition function T(c), where c is the number of courses: T(c) = Cost per course * c + Registration Fee T(c) = 35c + 375 The problem asks for the number of courses (c) where her tuition [I]cannot exceed[/I] $1000. The phrase [I]cannot exceed[/I] means less than or equal to, or no more than. So we setup the inequality for T(c) <= 1000 below: 35c + 375 <= 1000 To solve this inequality for c, we [URL=' it in our search engine and we get[/URL]: c <= 17.85 Since we cannot have fractional courses, we round down and get: c[B] <= 17[/B]

Jason decided that he will sell his stocks if their values per share (x) goes below $5 or above $15. Write a compound inequality represents the values at which Jason will sell his stocks? Below $5 is also known as less than $5: x < 5 Above $15 is also known as greater than $15 x > 15 We write the compound inequality: [B]x < 5 U x > 15[/B]

Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph. a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit? b. What proportion of the vehicles would be going less than 50 mph? c. A new speed limit will be initiated such that approximately 10% of vehicles will be over the speed limit. What is the new speed limit based on this criterion? d. In what way do you think the actual distribution of speeds differs from a normal distribution? a. Using our [URL=' calculator[/URL], we see that P(x<65) = [B]22.66%[/B] b. Using our [URL=' calculator[/URL], we see that P(x<50) = [B]0.4269%[/B] c. [URL=' of normal for 90% percentile[/URL] = 1.281551566 Plug into z-score formula: (x - 71)/8 = 1.281551566 [B]x = 81.25241252[/B] d. [B]The shape/ trail differ because the normal distribution is symmetric with relatively more values at the center. Where the actual has a flatter trail and could be expected to occur.[/B]

Barbra is buying plants for her garden. She notes that potato plants cost $3 each and corn plants cost $4 each. If she plans to spend at least $20 and purchase less than 15 plants in total, create a system of equations or inequalities that model the situation. Define the variables you use. [U]Define variables[/U] [LIST] [*]Let c be the number of corn plants [*]Let p be the number of potato plants [/LIST] Since cost = price * quantity, we're given two inequalities: [LIST=1] [*][B]3p + 4c >= 20 (the phrase [I]at least[/I] means greater than or equal to)[/B] [*][B]c + p < 15[/B] [/LIST]

Brighthouse charges $120 a month for their basic plan, plus $2.99 for each on demand movie you buy. Write and solve and inequality to find how many on demand movies could you buy if you want your bill to be less than $150 for the month. Let x equal to the number room movie rentals per month. Our inequality is: 120 + 2.99x < 150 To solve for the number of movies, Add 120 to each side 2.99x < 30 Divide each side by 2.99 x < 10.03, which means 10 since you cannot buy a fraction of a movie

Carmen has $30 in store bucks and a 25% discount coupon for a local department store. What maximum dollar amount can Carmen purchase so that after her store bucks and discount are applied, her total is no more than $60 before sales tax Let the original price be p. p Apply 25% discount first, which is the same as subtracting 0.25: p(1 - 0.25) Subtract 30 for in store buck p(1 - 0.25) - 30 The phrase [I]no more than[/I] means an inequality using less than or equal to: p(1 - 0.25) - 30 <= 60 To solve this inequality for p, we [URL=' it in our math engine[/URL] and we get: [B]p <= 120[/B]

Carmen is serving her child french fries and chicken wings for lunch today. Let f be the number of french fries in the lunch, and let c be the number of chicken wings. Each french fry has 25 calories, and each chicken wing has 100 calories. Carmen wants the total calorie count from the french fries and chicken wings to be less than 500 calories. Using the values and variables given, write an inequality describing this. We have: 25f + 100c < 50 Note: We use < and not <= because it states less than in the problem.

Chang is serving his child french fries and chicken wings for lunch today. Let f be the number of french fries in the lunch, and let c be the number of chicken wings. Each french fry has 25 calories, and each chicken wing has 100 calories. Chang wants the total calorie count from the french fries and chicken wings to be less than 600 calories. Using the values and variables given, write an inequality describing this. We have [B]25f + 100c < 600[/B] as our inequality.

Free Comparison of Numbers Calculator - Compares two numbers and checks to see if they are equal to one another, if the first number is greater than the second number, or the first number is less than the second number. Minimum and maximum.

Craig went bowling with $25 to spend. He rented shoes for $5.25 and paid $4.00 for each game. What was the greatest number of games Craig could have played? Set up the cost function C(g) where g is the number of games Craig plays: C(g) = Game fee * number of games (g) + shoe rental fee C(g) = 4g + 5.25 The problem asks for the maximum number of games Craig can play for $25. So we want an inequality of [I]less than or equal to[/I]. 4g + 5.25 <= 25 [URL=' this inequality into our search engine[/URL], and we get: g <= 4.9375 We want exact games, so we round this down to [B]4 games[/B].

Dawn has less than $60. She wants to buy 3 sweaters. What price of sweaters can she afford if all the sweaters are the same price? Let s be the price of each sweater. Write this as an inequality. The phrase [I]less than[/I] means an inequality, so we have the following inequality: 3s < 60 To solve this inequality, we [URL=' it in our math engine[/URL] and we get: s < [B]20[/B]

Emily is three years older than twice her sister Marys age. The sum of their ages is less than 30. What is the greatest age Mary could be? Let e = Emily's age and m = Mary's age. We have the equation e = 2m + 3 and the inequality e + m < 30 Substitute the equation for e into the inequality: 2m + 3 + m < 30 Add the m terms 3m + 3 < 30 Subtract 3 from each side of the inequality 3m < 27 Divide each side of the inequality by 3 to isolate m m < 9 Therefore, the [B]greatest age[/B] Mary could be is 8, since less than 9 [U]does not include[/U] 9.

This is a least common multiple problem. [URL=' least common multiple of 6 and 8 is 24[/URL] So every 24th person, less than or equal to 329 receives both a soda [U]and[/U] a hot dog. Using our multiples calculator, we find there are [URL=' multiples of 24 less than or equal to 329[/URL]. 24,48,72,96,120,144,168,192,216,240,264,288,312

Express the fact that x differs from 3 by less than 2/7 as an inequality involving absolute value. Solve for x. Let's build this algebraic expression in pieces: The phrase [I]differs from[/I] means a difference. x - 3 By less than 2/7 means we use the < sign compared to 2/7 x - 3 < 2/7 Finally, the problem says we involve absolute value. So we write this as: [B]|x - 3| < 2/7[/B]

Five less than a number is at least -7 and at most 7. A number signifies an arbitrary variable, let's call it x. Five less than a number: x - 5 Is at least -7 means greater than or equal to and at most 7 means less than or equal to, so we have a joint inequality: [B]-7 <= x - 5 <= 7[/B]

For her phone service, Maya pays a monthly fee of $27 , and she pays an additional $0.04 per minute of use. The least she has been charged in a month is $86.04 . What are the possible numbers of minutes she has used her phone in a month? Use m for the number of minutes, and solve your inequality for m . Maya's cost function is C(m), where m is the number of minutes used. C(m) = 0.04m + 27 We are given C(m) = $86.04. We want her cost function [I]less than or equal[/I] to this. 0.04m + 27 <= 86.04 [URL=' this inequality into our search engine[/URL], and we get [B]m <= 1476[/B].

Free Interval Notation and Set Builder Notation Calculator - This calculator translates the following inequality statements to interval notation and set builder notation:
* x < 5
* y <= 5
* z > 5
* a >= 5
* b < 5 or b > 20
* Compound Inequality such as 0 <= c < 4
* |x|<3
* Reverse Interval Notation to Inequality Statement such as (-7,5]
* {x|x<1}
* Word representations of interval notations such as 2 is less than or equal to x is less than or equal to 8

jane has 55$ to spend at cedar point. the admission price is 42$ and each soda is 4.25. write an inequality to show how many sodas he can buy. Let s be the number of sodas. Cost for the day is: Price per soda * s + Admission Price 4.25s + 42 We're told that Jane has 55, which means Jane cannot spend more than 55. Jane can spend up to or less than 55. We write this as an inequality using <= 55 [B]4.25s + 42 <= 55[/B] [B][/B] If the problems asks you to solve for s, we type it in our math engine and we get: Solve for [I]s[/I] in the inequality 4.25s + 42 ? 55 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 42 and 55. To do that, we subtract 42 from both sides 4.25s + 42 - 42 ? 55 - 42 [SIZE=5][B]Step 2: Cancel 42 on the left side:[/B][/SIZE] 4.25s ? 13 [SIZE=5][B]Step 3: Divide each side of the inequality by 4.25[/B][/SIZE] 4.25s/4.25 ? 13/4.25 [B]s ? 3.06[/B]

Jennifer is playing cards with her bestie when she draws a card from a pack of 25 cards numbered from 1 to 25. What is the probability of drawing a number that is square? The squares from 1 - 25 less than or equal to 25 are as follows: [LIST=1] [*]1^2 = 1 [*]2^2 = 4 [*]3^2 = 9 [*]4^2 = 16 [*]5^2 = 25 [/LIST] So the following 5 cards are squares: {1, 4, 9, 16, 25} Therefore, our probability of drawing a square is: P(square) = Number of Squares / Number of Cards P(square) = 5/25 This fraction can be simplified. So [URL=' type in 5/25 into our search engine, choose simplify[/URL], and we get: P(square) = [B]1/5[/B]

Joey and Romnick play in the same soccer team. Last Saturday, Romnick scored 3 more goals than Joey,but together they scored less than 9 goals. What are the possible number of goal Romnick scored? Let j be Joey's goals Let r by Romnick's goals We're given 1 equation and 1 inequality: [LIST=1] [*]r = j + 3 [*]r + j < 9 [/LIST] Rearranging equation 1 for j, we have: [LIST=1] [*]j = r - 3 [*]r + j < 9 [/LIST] Substitute equation (1) into inequality (2) for j: r + r - 3 < 9 2r - 3 < 9 [URL=' this inequality into our math engine[/URL], we get: [B]r < 6[/B]

JP's age is twice the age of Reyna. The sum of their ages does not exceed 51 Let JP's age be j. Let Reyna's age be r. We're given two expressions: [LIST=1] [*]w = 2r [*]r + w <= 51. ([I]Does not exceed means less than or equal to)[/I] [/LIST] We substitute (1) into (2) for w to get the inequality: r + 2r <= 51 To solve this inequality, we type it in our search engine and we get: [B]r <= 17[/B]

Juan has d dimes and q quarters in his pocket. The total value of the coins is less than $14.75. Which inequality models this situation? Since dimes are worth $0.10 and quarters are worth $0.25, we have: [B]0.10d + 0.25q < 14.75[/B]

Juan spent at most $2.50 on apples and oranges. He bought 5 apples at $0.36 each. What is the most he spent on oranges? Let a be spending apples and o be spending on oranges, we have: [LIST=1] [*]a + o <= 2.36 <-- At most means less than or equal to [*]a = 5 * 0.36 = 1.8 [/LIST] Substitute (2) into (1) 1.8 + o <= 2.36 Subtract 1.8 from each side [B]o <= 0.56[/B]

Karen bought a bucket of popcorn at the movies for $5. She also bought some candy for $2 each. Karen has to spend less than $15 on the popcorn and candy. Which inequality can be used to find c, the number of candies that Karen could have bought? Since the candy cost is the product of price and quantity, we have: 2c + 5 < 15 To solve this inequality for c, we [URL=' it in our math engine [/URL]and we get: [B]c < 5[/B]

Kate spent at most $2.50 on apples and oranges. She bought 5 apples at $0.36 each. What is the most She spent on oranges [U]Assumptions and givens:[/U] [LIST] [*]Let a be the total cost of apples [*]Let o be the total cost of oranges [/LIST] The phrase [I]at most[/I] means less than or equal to, so we have: a + o <= 2.50 [U]Find the cost of apples (a)[/U] a = price per apple * quantity of apples a = 0.36 * 5 a = 1.8 Our new inequality with a = 1.8 is: 1.8 + o <= 2.50 [URL=' this inequality into our search engine[/URL], we get: [B]o <= 0.7[/B]

Keith is going to Renaissance Festival with $120 to pay for his admission, food and the cost of games. He spends a total of $85 on admission and food. Games cost $5 each. Which inequality models the maximum number of games Keith can play. Let the number of games be g. Keith can spend less than or equal to 120. So we have [B]5g + 85 <= 120 [/B] If we want to solve the inequality for g, we [URL=' it in our search engine[/URL] and we have: g <= 7

kim wants to buy candy for 4 dollars a pound. if she wants to spend less than 20 dollars, how many bags can she buy Since cost = price * quantity, we have the following inequality with b as the number of bags: 4b < 20 To solve this inequality for b, we [URL=' it in our search engine[/URL] and we get: [B]b < 5[/B]

kira will spend less than 27 on gifts. so far, she has spent 12$. what are the possible additional amounts she will spend? The key word in this problem is [I]less than[/I]. So we know this is an inequality. Let s be Kira's possible spend. We have: s + 12 < 27 To solve for s in this inequality, we subtract 12 from each side: s + 12 - 12 < 27 - 12 Cancel the 12's on the left side, and we get: [B]s < 15 [/B] [I]The summary here is Kira can spend anything up to [U]but not including[/U] 15[/I]

Lisa wants to rent a boat and spend less than $52. The boat costs $7 per hour, and Lisa has a discount coupon for $4 off. What are the possible numbers of hours Lisa could rent the boat? Calculate discounted cost: Discounted cost = Full Cost - Coupon Discounted cost = 52 - 7 Discounted cost = 45 Since price equals rate * hours (h), and we want the inequality (less than) we have: 7h < 52 Using our [URL=' calculator,[/URL] we see that: [B]h < 7.42[/B]

M is the set of integers that are greater than or equal to -1 and less than or equal to 2 We include -1 on the left, and include 2 on the right [B]M = {-1, 0, 1, 1, 2)[/B]

Marcela is having a presidential debate watching party with all of her friends, She will be making chicken wings and hot dogs. Each chicken wing costs $2 to make and each hot dog costs $3. She needs to spend at least $500. Marcela knows that she will make more than 50 chicken wings and hot dogs combined. She also knows that she will make less than 120 chicken wings and less that 100 hot dogs. What are her inequalities? Let c be the number of chicken wings and h be the number of hot dogs. Set up the given inequalities: [LIST=1] [*]c + h > 50 [I]Marcela knows that she will make more than 50 chicken wings and hot dogs combined.[/I] [*]2c + 3h >= 500 [I]She needs to spend at least $500[/I] [*]c < 120 [I]She also knows that she will make less than 120 chicken wings[/I] [*]h < 100 [I]and less that 100 hot dogs[/I] [/LIST]

Mr. Smith wants to spend less than $125 at a zoo. A ticket cost $7 he is taking 2 kids with him. Use p to represent the other money he can spend there. 2 kids and Mr. Smith = 3 people. Total Ticket Cost is 3 people * 7 per ticket = 21 If he has 125 to spend, we have the following inequality using less than or equal to (<=) since he can spend up to or less than 125: p + 21 <= 125 Subtract 21 from each side: [B]p <= 104[/B]

Nine less than a number is no more than 8 and no less than 3 A number is denoted as an arbitrary variable, let's call it x. We have a double inequality: [LIST=1] [*]No more than 8 means less than or equal to 8 [*]No less than 3 means greater than or equal to 3 [/LIST] [B]3 <= x <= 8[/B]

The product of 2 and y means we multiply 2y Nine less than that product means we subtract 9 2y - 9 Finally, the entire expression is not less than 15, which means 15 or more. We use greater than or equal to [B]2y - 9 >= 15 [/B] If you want to solve this inequality and write the interval notation, use [URL=' calculator[/URL].

People with a drivers license are at least 16 years old and no older than 85 years old. Set up the inequality, where p represents the people: [LIST=1] [*]The phrase [I]at least[/I] means greater than or equal to. So we use the >= sign. 16 <= p [*]The phrase [I]no older than[/I] means less than or equal to. So we use the <= sign. p <= 85 [/LIST] Combine these inequalities, and we get: [B]16 <= p <= 85[/B] To see the interval notation for this inequality and all possible values, visit the [URL=' notation calculator[/URL].

Pet supplies makes a profit of $5.50 per bag, if the store wants to make a profit of no less than $5225, how many bags does it need to sell? 5.5ob >= $5,225 Divide each side of the inequality by $5.50 b >=9.5 bags, so round up to a whole number of 10 bags.

Update, this has been added to our shortcuts. You can type any expression in the form, positive numbers less than x where x is any integer. You can also type positive numbers greater than x where x is any integer. Same with less than or equal to and greater than or equal to.

r less than 164 is 248 more than the product of 216 and r [U]r less than 164:[/U] 164 - r [U]The product of 216 and r:[/U] 216r [U]248 more than the product of 216 and r[/U] 216r + 248 [I]The word is means an equation, so we set 164 - r equal to 216r + 248[/I] [B]164 - r = 216r + 248[/B]

Rhonda raised $245 for her softball team's fundraiser.She wants to raise no less than $455.Write and solve an inequality to determine how much more money Rhonda must raise to reach her goal. Let d represent the amount of money in dollars Rhonda must raise to reach her goal. The phrase [I]no less than[/I] is an inequality using the greater than or equal sign: d >= 455 - 245 d >= [B]210[/B]

Rita has spent $16 so far on gifts. The additional amount she will spend will be less than $14. What are the possible total amounts she will spend? Rita will spend at least another cent on other gifts above the $16 she spent so far, but no more than $14. Also, the problem says less than 14. 16 + 14 is 30, so that is the top end of her spending. Let's say her remaining spending is s. Set up the inequality for possible spending values. [B]16 < s < 30[/B]

C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} Notice, we do not include 20 since it said less than, and not less than or equal to.

Seven less than 1/4 of a number is 9. The phrase [I]a number[/I] means an arbitrary variable, let's call it x. x 1/4 of a number means we multiply x by 1/4: x/4 Seven less than this means we subtract 7 from x/4: x/4 - 7 The word [I]is[/I] means an equation, so we set x/4 - 7 equal to 9: [B]x/4 - 7 = 9[/B]

First, the phrase [I]a number[/I] means we choose an arbitrary variable, let's call it x. Twice a number means we multiply it by 2. 2x Six less than that means we subtract 6 2x - 6 Now, the last piece, we set up an inequality. At least -1 means greater than or equal to 1. At most 1 means less than or equal to 1. Notice, for both points, we include the number. -1 <= 2x - 6 <= 1

The auditorium can hold a maximum of 150 people We want an inequality for the number of people (p) in the auditorium. The word [I]maximum[/I] means [I]no more than[/I] or [I]less than or equal to[/I]. So we have: [B]p <= 150[/B]

The base of a triangle with a height of 7 units is represented by the formula b=2/7A. The base of the triangle is less than 10 units. Write and solve an inequality that represents the possible area A of the triangle We're given: b=2/7A We're also told that b is less than 10. So we have: 2/7A < 10 2A/7 < 10 Cross multiply: 2A < 7 * 10 2A < 70 Divide each side of the inequality by 2 to isolate A 2A/2 < 70/2 Cancel the 2's on the left side and we get: A < [B]35[/B]

The cost of 25 apples is less than $9.50. The cost of 12 apples is more than 3.60. What are the possible prices of one apple? Let a be the price of each apple. We're given 2 inequalities: [LIST=1] [*]25a < 9.50 [*]12a > 3.60 [/LIST] [URL=' 25a < 9.50 into our search engine[/URL], we get a < 0.38 [URL=' 12a > 360 into our search engine[/URL], we get a > 0.3 Therefore, the possible prices a of one apple are expressed as the inequality: [B]0.3 < a < 0.38[/B]

The cost of renting a rototiller is $19.50 for the first hour and $7.95 for each additional hour. How long can a person have the rototiller if the cost must be less than $95? Setup the inequality: $19.50 + $7.95x < $95 Subtract 19.50 from both sides: 7.95x < 75.50 Divide each side of the inequality by 7.95 to isolate x x < 9.5 The next lowest integer is 9. So we take 9 + the first hour of renting to get [B]10 total hours[/B]. Check our work: $7.95 * 9.5 + $19.50 $71.55 + $19.50 = $91.05

The difference between A and B is no less than 30 The difference between means we subtract. No less than means greater than or equal to, so we have the following inequality; [B]A - B >= 30[/B]

the difference between A and B is no less than 30. The difference between a and b: a - b The phrase [I]no less than[/I] means an inequality. You can also say this as [I]greater than or equal to[/I]. [B]a - b >= 30[/B]

The difference of twice a number and 6 is at most 28 This is an algebraic expression. Let's take it in parts: [LIST=1] [*]The phrase [I]a number[/I], means an arbitrary variable, let's call it x [*]Twice this number means we multiply x by 2: 2x [*][I]The difference of[/I] means subtract, so we subtract 6 to 2x: 2x - 6 [*][I]Is at most [/I]means less than or equal to, so we create an inequality where 2x - 6 is less than or equal to 28, using the <= sign [/LIST] [B]2x - 6 <= 28 [/B] If you wish to solve this inequality, [URL=' this link[/URL].

the difference of twice a number and 8 is at most -30. The phrase [I]a number[/I] means an arbitrary variable, let's call it x. Twice this number means we multiply by 2, so we have 2x. We take the difference of 2x and 8, meaning we subtract 8: 2x - 8 Finally, the phrase [I]at most[/I] means an inequality, also known as less than or equal to: [B]2x - 8 <= 30 <-- This is our algebraic expression [/B] To solve this, we [URL=' it into the search engine[/URL] and get x <= 19.

The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered that first digits do not occur with equal frequency. Probabilities of occurrence to the first digit in a number are shown in the accompanying table. The probability distribution is now known as Benford's Law. For example, the following distribution represents the first digits in 231 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer. Digit, Probability 1, 0.301 2, 0.176 3, 0.125 4, 0.097 5, 0.079 6, 0.067 7, 0.058 8, 0.051 9, 0.046 [B][U]Fradulent Checks[/U][/B] Digit, Frequency 1, 36 2, 32 3, 45 4, 20 5, 24 6, 36 7, 15 8, 16 9, 7 Complete parts (a) and (b). (a) Using the level of significance α = 0.05, test whether the first digits in the allegedly fraudulent checks obey Benford's Law. Do the first digits obey the Benford's Law?
Yes or No Based on the results of part (a), could one think that the employe is guilty of embezzlement? Yes or No Show frequency percentages Digit Fraud Probability Benford Probability 1 0.156 0.301 2 0.139 0.176 3 0.195 0.125 4 0.087 0.097 5 0.104 0.079 6 0.156 0.067 7 0.065 0.058 8 0.069 0.051 9 0.03 0.046 Take the difference between the 2 values, divide it by the Benford's Value. Sum up the squares to get the Test Stat of 2.725281277 Critical Value Excel: =CHIINV(0.95,8) = 2.733 Since test stat is less than critical value, we cannot reject, so [B]YES[/B], it does obey Benford's Law and [B]NO[/B], there is not enough evidence to suggest the employee is guilty of embezzlement.

The price of a baseball glove is no more than $38.95. Let p be the price of the baseball glove. The phrase "no more than" means less than or equal to. Our inequality is: p <= $38.95

The price of a cheap backpack is $15 less than an expensive backpack. When Emily bought both, she paid $75. What is the cost of the cheap backpack? backpack cost = b Cheap backpack = b - 15 The total of both items equals 75: b + b - 15 = 75 Solve for [I]b[/I] in the equation b + b - 15 = 75 [SIZE=5][B]Step 1: Group the b terms on the left hand side:[/B][/SIZE] (1 + 1)b = 2b [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 2b - 15 = + 75 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants -15 and 75. To do that, we add 15 to both sides 2b - 15 + 15 = 75 + 15 [SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE] 2b = 90 [SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE] 2b/2 = 90/2 b = 45 Cheap backpack = 45 - 15 = [B]30 [URL='

Let's take this by pieces. The product of 8 and a number k is written as: 8k. Since it's greater than 4, but not more than 16, we include this in the middle of an inequality statement. 4 < 8k <= 16 Notice no more than has an equal sign, it means less than or equal to. Greater does not include an equal sign.

the product of a number and 15 is not less than 15 The phrase [I]a number[/I] means an arbitrary variable. Let's call it x. x the product of a number and 15 means we multiply x by 15 15x The phrase [I]not less than[/I] means greater than or equal to. We set 15x greater than prequel to 15 [B]15x >= 15 <-- This is our algebraic expression [/B] [U]If the problem asks you to solve for x:[/U] Divide each side by 15: 15x/15 >= 15/15 [B]x >= 1[/B]

The product of a number b and 3 is no less than 12. A number b is just written as b. So we have: The product of b and 3 is no less than 12. take this in parts: [LIST] [*]The product of b and 3: 3b [*]The phrase [I]is no less than[/I] means an inequality, so we have greater than or equal to. We set 3b greater than or equal to 12 [/LIST] [B]3b >= 12[/B]

The product of x and 7 is not greater than 21 The product of x and 7: 7x Is not greater than means less than or equal to, so we have our algebraic expression: 7x <= 21 If you want to solve this inequality and interval notation, use our [URL='

The product of x and u is not greater than 21 The product of x and u xu Not greater than means less than or equal to: xu <= 21

The sum of 2 and w is less than or equal to 27. Take this algebraic expression in parts: [LIST] [*]The sum of 2 and w: 2 + w [*]The phrase [I]less than or equal to[/I] means an inequality, using the <= sign. [/LIST] [B]2 + w <= 27[/B]

The sum of 5 and 2x is at most 27 The sum of 5 and 2x means we add 2x to 5: 5 + 2x The phrase [I]at most[/I] means less than or equal to, so we have an inequality where 5 + 2x is less than or equal to 27 [B]5 + 2x <= 27[/B]

the sum of 5 and y is less than or equal to -21 Take this algebraic expression in parts: The sum of 5 and y means we add y to 5 5 + y The phrase [I]less than or equal to[/I] -21 means an inequality. We use the <= sign to relate 5 + y to -21 [B]5 + y <= -21[/B]

The sum of a number b and 3 is greater than 4 and no more than 16 The sum of a number b and 3: b + 3 Greater than 4 and no more than 16 means we have a combo inequality: [LIST] [*]Greater than 4 means we use a > sign [*]No more than 16 means less than or equal to, so <= [/LIST] [B]4 < b + 3 <= 16[/B]

the sum of a number times 3 and 30 is less than 17 A number is denoted as an arbitrary variable, let's call it x. x Times 3 means we multiply x by 3: 3x The sum of a number times 3 and 30 means we add 30 to 3x above 3x + 30 Is less than 17 means we have an inequality, so we set 3x + 30 less than 17 3x + 30 < 17 To solve for x and see the interval notation, use [URL=' calculator[/URL]:

The sum of x and y is at most 10 The sum of x and y: x + y Is at most 10 means we have an inequality, at most means 10 or less, so less than or equal to [B]x + y <= 10[/B]

The temperature inside the lab refrigerator is no more than 35 . Use t to represent the temperature (in ) of the refrigerator. The phrase [I]no more than[/I] means less than or equal to. We have this inequality: [B]t <= 35[/B]

The width of a rectangle is fixed at 4cm. For what lengths will the area be less than 86 cm^2 The Area (A) of a rectangle is given by: A = lw With an area of [I]less than[/I] 86 and a width of 4, we have the following inequality: 4l < 86 To solve for l, we [URL=' this inequality into our search engine[/URL] and we get: [B]l < 21.5[/B]

[SIZE=4]There are 5 pencil-cases on the desk. Each pencil-case contains at least 10 pencils, but not more than 14 pencils. Which of the following could be the total number of pencils in all 5 cases? A) 35 B) 45 C) 65 D) 75 [U]Determine the minimum amount of pencils (At least means greater than or equal to):[/U] Minimum Amount of pencils = Cases * Min Quantity Minimum Amount of pencils = 5 * 10 Minimum Amount of pencils = 50 [SIZE=4][U]Determine the maximum amount of pencils (Not more than means less than or equal to):[/U] Maximum Amount of pencils = Cases * Min Quantity Maximum Amount of pencils = 5 * 14 Maximum Amount of pencils = 70[/SIZE] So our range of pencils (p) is: 50 <= p <= 70 Now take a look at our answer choices. The only answer which fits in this inequality range is [B]C, 65[/B]. [B][/B][/SIZE]

This is a double inequality. Let's take it by pieces: Three more than 2x is denoted as 2x + 3. We add since we see the phrase, [I]more than[/I]. Because it's greater than or equal to 1, we have: 1 <= 2x + 3 Finally, that same phrase is [U]also[/U] less than or equal to 11. 2x + 3 <= 11. Piecing these two inequalities together, we have: 1 <= 2x + 3 <= 11.

This is a two-part inequality. Let's take it piece by piece. Three more than 2x means we add. 2x + 3 It's greater than or equal to 1, denoted below: 1 <= 2x + 3 It's also less than or equal to 11, denoted below 2x + 3 <= 11 Piece these two inequalities together: 1 <= 2x + 3 <= 11

Twice x means we multiply x by 2: 2x five less than twice x 2x - 5 Three x 3x The word [I]is[/I] means equal to. Set 2x - 5 equal to 3x for our algebraic expression: [B]2x - 5 = 3x [/B] If the problem asks you to solve for x, subtract 2x from each side [B]x = -5[/B]

two thirds of a number is no more than -10 The phrase [I]a number[/I] means an arbitrary variable, let's call it x: x Two thirds of a number mean we multiply x by 2/3: 2x/3 The phrase [I]no more than[/I] -10 means less than or equal to -10, so we have an inequality: [B]2x/3 <= -10[/B]

Victoria is 4 years older than her neighbor. The sum of their ages is no more than 14 years. Let Victoria's age be v. And her neighbor's age be n. We're given: [LIST=1] [*]v = n + 4 [*]v + n <=14 <-- no more than means less than or equal to [/LIST] Substitute Equation (1) into Inequality (2): (n + 4) + n <= 14 Combine like terms: 2n + 4 <= 14 [URL=' this inequality into our search engine[/URL], we get: n <= 5 Substituting this into inequality (2): v + 5 <= 14 [URL=' this inequality into our search engine[/URL], we get: [B]v <= 9[/B]

Video store movie rental plans. Plan A 25 membership fee plus 1.25 for movie. Plan B 40 for unlimited rentals. What number of movies rentals is plan B less than plan A? Let x equal the number of movies rented and C the cost for rentals Plan A: C = 1.25x + 25 Plan B: C = 40 Set up the inequality: 1.25x + 25 > 40 Subtract 25 from each side: 1.25x > 15 Divide each side of the inequality by 1.25 x > 12 So [B]13[/B] rentals or more make Plan B less than Plan A.

X is the speed limit is a maximum 65 mph A maximum of means less than or equal to. Or, no more than. So we have the inequality: [B]X <= 65[/B]

X to the 9th is less than or equal to 38: x to the 9th means 9th power: x^9 We set this less than or equal 38: [B]x^9 <= 38[/B]

Yolanda wants to rent a boat and spend less than $41. The boat costs $8 per hour, and Yolanda has a discount coupon for $7 off. What are the possible numbers of hours Yolanda could rent the boat? A few things to build this problem: [LIST=1] [*]Discount subtracts from our total [*]Cost = Hourly rate * hours [*]Less than means an inequality using the < sign [/LIST] Our inequality is: 8h - 7 < 41 To solve this inequality for h, we [URL=' it in our math engine[/URL] and we get: h < [B]6[/B]

You can spend at most $35. If you buy 5 tickets, how much can you spend on each ticket We're given the number of tickets as 5. We know cost = price * quantity Let p = price The phrase [B]at most[/B] means less than or equal to, so we have: 5p <= 35 [URL=' this inequality into our search engine[/URL], we have: [B]p <= 7[/B]

You have $6.50 to make copies. It cost $0.45. Write and solve an equality that represents the number of copies Hoow many exact copies can you make? Let the number of copies be c. We have: 0.45c = 6.50 [URL=' this equation into our search engine[/URL] and we get: c = 14.444 We round down and say we can make 14 copies. [B]c = 14[/B] Now, if the problem asks you for an [I]inequality[/I], we want to see how many copies we can make without exceeding our $6.50 spend. So it's less than or equal to: [B]c <= 14[/B]