7.EE.4b: Solve word problems leading to inequalities of the form px...

Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.



Jonathan wants to save up enough money so that he can buy a new sports equipment set that includes a football, baseball, soccer ball, and basketball. This complete boxed set costs $50. Jonathan has $15 he saved from his birthday. In order to make more money, he plans to wash neighbors’ windows. He plans to charge $3 for each window he washes, and any extra money he makes beyond $50 he can use to buy the additional accessories that go with the sports box set.

Write and solve an inequality that represents the number of windows Jonathan can wash in order to save at least the minimum amount he needs to buy the boxed set. Graph the solutions on the number line. What is a realistic number of windows for Jonathan to wash? How would that be reflected in the graph?
Jonathan wants to save up enough money so that he can buy a new sports equipment set that includes a football, baseball, soccer ball, and basketball. This complete boxed set costs $50. Jonathan has $15 he saved from his birthday. In order to make more money, he plans to wash neighbors’ windows. He plans to charge $3 for each window he washes, and any extra money he makes beyond $50 he can use to buy the additional accessories that go with the sports box set.

Write and solve an inequality that represents the number of windows Jonathan can wash in order to save at least the minimum amount he needs to buy the boxed set. Graph the solutions on the number line. What is a realistic number of windows for Jonathan to wash? How would that be reflected in the graph?


Chandler was given $75 for a birthday present. This present, along with earnings from a summer job, is being set aside for a mountain bike. The job pays $6 per hour, and the bike costs $345. To be able to buy the bike, how many hours does Chandler need to work?

(Continuation) Let h be the number of hours that Chandler works. What quantity is represented by the expression 6h? What quantity is represented by the expression 6h+75?

(a) Graph the solutions to the inequality \(6h+75\ge345\) on a number line.
(b) Graph the solutions to the inequality \(6h+75< 345\) on a number line.
(c) What do the solutions to the inequality \(6h+75\ge345\) signify?
Chandler was given $75 for a birthday present. This present, along with earnings from a summer job, is being set aside for a mountain bike. The job pays $6 per hour, and the bike costs $345. To be able to buy the bike, how many hours does Chandler need to work?

(Continuation) Let h be the number of hours that Chandler works. What quantity is represented by the expression 6h? What quantity is represented by the expression 6h+75?

(a) Graph the solutions to the inequality \(6h+75\ge345\) on a number line.
(b) Graph the solutions to the inequality \(6h+75< 345\) on a number line.
(c) What do the solutions to the inequality \(6h+75\ge345\) signify?
Phillips Exeter Academy (no answers available)  ■  edit


Eugene and Wes are solving the inequality \(132-4x\le36\). Each begins by subtracting 132 from both sides to get \(-4x\le-96\), and then each divides both sides by-4. Eugene gets \(x\le24\) and Wes gets \(x\ge24\), however. Always happy to offer advice, Alex now suggests to Eugene and Wes that answers to inequalities can often be checked by substituting \(x=0\) into both the original inequality and the answer. What do you think of this advice? Graph each of these answers on a number line. \linebreak


(Continuation) After hearing Alex’s suggestion about using a test value to check an in- equality, Cameron suggests that the problem could have been done by solving the equation \(132-4x=36\) first. Complete the reasoning behind this strategy.\linebreak


(Continuation) Deniz, who has been keeping quiet during the discussion, remarks, “The only really tricky thing about inequalities is when you try to multiply them or divide them by negative numbers, but this kind of step can be avoided altogether. Cameron just told us one way to avoid it, and there is another way, too.” Explain this remark by Deniz.\linebreak


Eugene and Wes are solving the inequality \(132-4x\le36\). Each begins by subtracting 132 from both sides to get \(-4x\le-96\), and then each divides both sides by-4. Eugene gets \(x\le24\) and Wes gets \(x\ge24\), however. Always happy to offer advice, Alex now suggests to Eugene and Wes that answers to inequalities can often be checked by substituting \(x=0\) into both the original inequality and the answer. What do you think of this advice? Graph each of these answers on a number line. \linebreak


(Continuation) After hearing Alex’s suggestion about using a test value to check an in- equality, Cameron suggests that the problem could have been done by solving the equation \(132-4x=36\) first. Complete the reasoning behind this strategy.\linebreak


(Continuation) Deniz, who has been keeping quiet during the discussion, remarks, “The only really tricky thing about inequalities is when you try to multiply them or divide them by negative numbers, but this kind of step can be avoided altogether. Cameron just told us one way to avoid it, and there is another way, too.” Explain this remark by Deniz.\linebreak

Phillips Exeter Academy (no answers available)  ■  edit
Solve two-step linear inequalities



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