3-3 Solving Inequalities Using Multiplication or Division

Quick Review

You can use the multiplication and division properties of inequality to transform an inequality. When you multiply or divide each side of an inequality by a negative number, you have to reverse the inequality symbol.

Example

What are the solutions of − 3 x > 12 ? negative 3 bold x greater than 12 question mark

− 3 x > 12   − 3 x − 3 < 12 − 3 Divide each side by  − 3.  Reverse the inequality symbol . x < − 4 Simplify . table with 3 rows and 3 columns , row1 column 1 , negative 3 x , column 2 greater than 12 , column 3 , row2 column 1 , fraction negative 3 x , over negative 3 end fraction , column 2 less than , 12 over negative 3 , column 3 table with 2 rows and 1 column , row1 column 1 , cap divideeachsideby . minus 3. . cap reversethe , row2 column 1 , inequalitysymbol . . , end table , row3 column 1 , x , column 2 less than negative 4 , column 3 cap simplify , . , end table

Exercises

Solve each inequality. Graph your solutions.

20. 5 x < 15 5 x less than 15
21. − 6 t > 18 negative 6 t greater than 18
22. y 3 ≤ 2 y over 3 , less than or equal to 2
23. − h 4 < 6 negative , h over 4 , less than 6
24. 25.5 g > 102 25.5 , g greater than 102
25. − 3 5 n ≥ − 9 negative , 3 fifths , n greater than or equal to negative 9
26. 44.5 ≤ 2.7 d 44.5 , less than or equal to 2.7 d
27. − 17.1 m < 23.8 negative , 17.1 , m less than , 23.8
28. Part-Time Job You earn $7.25 per hour baby-sitting. Write and solve an inequality to find how many full hours you must work to earn at least $200.

3-4 Solving Multi-Step Inequalities

Quick Review

When you solve inequalities, sometimes you need to use more than one step. You need to gather the variable terms on one side of the inequality and the constant terms on the other side.

Example

What are the solutions of 3 x + 5 > − 1 ? 3 bold x plus 5 greater than negative 1 question mark

3 x + 5 > − 1   3 x > − 6 Subtract  5  from each side  . x > − 2 Divide each side by  3 . table with 3 rows and 3 columns , row1 column 1 , 3 x plus 5 , column 2 greater than negative 1 , column 3 , row2 column 1 , 3 x , column 2 greater than negative 6 , column 3 cap subtract . 5 . fromeachside . . , row3 column 1 , x , column 2 greater than negative 2 , column 3 cap divideeachsideby . 3 . , end table

Exercises

Solve each inequality.

29. 4 k − 1 ≥ − 3 4 k minus 1 greater than or equal to negative 3
30. 6 ( c − 1 ) < − 18 6 open c minus 1 close less than negative 18
31. 3 t > 5 t + 12 3 t greater than 5 t plus 12
32. − 6 7 y − 6 ≥ 42 negative , 6 sevenths , y minus 6 greater than or equal to 42
33. 4 + x 2 > 2 x 4 plus , x over 2 , greater than 2 x
34. 3 x + 5 ≤ 2 x − 8 3 x plus 5 less than or equal to 2 x minus 8
35. 13.5 a + 7.4 ≤ 85.7 13.5 , eh plus 7.4 less than or equal to , 85.7
36. 42 w > 2 ( w + 7 ) 42 w greater than 2 open w plus 7 close
37. Commission A salesperson earns $200 per week plus a commission equal to 4% of her sales. This week her goal is to earn no less than $450. Write and solve an inequality to find the amount of sales she must have to reach her goal.

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