CONFIDENTIAL 1 Algebra1 Solving Inequalities by Multiplying or Dividing

CONFIDENTIAL 1

Algebra1Algebra1

Solving Inequalities Solving Inequalities by Multiplying or by Multiplying or

DividingDividing

CONFIDENTIAL 2

Warm UpWarm Up

Solve each inequality and graph the solutions.

1) n - 15 < 3

2) m - 13 > 29

3) v – 4 < 7

4) t – 5 > 11

1) n < 18

2) m > 42

3) n < 11

4) t > 16

CONFIDENTIAL 3

Inequalities

Solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division,

undo the operation by dividing or multiplying both sides of the inequality by the same number.

STEP1: Identify the variable.

STEP2: To get the variable by itself, Multiply the same number to or Divide the same number from each side of the inequality .

STEP3: Check the solution .

Solving inequality by Multiply or Divide needs certain steps to be followed.

CONFIDENTIAL 4

The rules are similar for a ≥ b and a ≤ b.

Multiplication property of inequality

When you multiply each side of a true inequality by a positive integer, the result remains true.

In symbol: For all integers a, b, and c, where c > 0.

1. If a > b, then a × c > b × c and

2. If 7 > 2, then 7 × 4 > 2 × 4

Multiplication and Division by Positive Numbers

CONFIDENTIAL 5

Division property of inequality


When you divide each side of a true inequality by a positive integer, the result remains true.


1. If a > b, then a > b, c c

2. If 7 > 3, then 7 > 3, 4 4

CONFIDENTIAL 6

Multiplying or Dividing by a Positive Number


A) 3x > -27

3x > -27 3 3

1x > -9

x > -9

Since x is multiplied by 3, divide both sides by 3 to undo the multiplication.

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x > -9

CONFIDENTIAL 7

B) 2r < 6 3

r < 9

Since r is multiplied by 2/3 , multiply both sides by the reciprocal of 2/3.

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

r < 9

2r < (6) 3

32

32

2r < 6 3

CONFIDENTIAL 8


Now you try!

1) 4k > 24 2) g > 2734

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

k > 6

1)

37 38 39 40 41 4231 32 33 34 35 36

g > 36

2)

43 44

CONFIDENTIAL 9

If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality

symbol to make the statement true.

5 > -3 5 is greater than -3.

Multiply both sides by -2.

You know that -10 is less than 6, so use the symbol for less than.

5 (-2) -3 (-2)

-10 6

-5 < 3

CONFIDENTIAL 10

Multiplying both sides by a negative number changes the sign of both sides of the inequality.

This means there is another set of properties of inequality for multiplying or dividing by a negative number.

0 2 4 6 8 1012 10 8 6 4 2

5

12 14

3

x(-2)

x(-2)

CONFIDENTIAL 11

The rules are similar for a > b and a < b.

When you multiply or divide each side of an inequality by a negative integer, you must reverse the order symbol.

In symbol: For all integers a, b, and c, where c < 0.

1. If a > b, then a x c < b x c and

2. If 7 > 3, then 7 x -5 < 3 x -5


Multiplication and Division by Negative Numbers

CONFIDENTIAL 12




1. If a > b, then a < b and c c

2. If 7 > 2, then 7 < 2, -4 -4



CONFIDENTIAL 13

A) -8x > 72

-8x > 72

x < -9

Since x is multiplied by -8, divide both sides by -8. Change > to <.

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x < -9

Multiplying or Dividing by a Negative Number


-8x < 72 -8 -8

CONFIDENTIAL 14

B) -3 ≤ x -5

(-5) -3 ≥ (-5)x -5

15 ≥ x (or x ≤ 15)

Since x is divided by -5, multiply both sides by -5. Change ≤ to ≥.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 18 19-3 -2 -1

x ≤ 15

20 21

CONFIDENTIAL 15


Now you try!

1) 10 ≥ -2x 2) 4.25 > -0.25h

1) x ≥ -5 2) h > -17

CONFIDENTIAL 16

Problem Solving Application

Ryan has a $16 gift card for a health store where a smoothie costs $2.50 with tax. What are the possible

numbers of smoothies that Ryan can buy?

Let s represent the number of smoothies Ryan can buy.

$2.50 times number of smoothies is at most $16.00.

2.50 • s ≤ 16.00

s ≤ 6.4

2.50 • s ≤ 16.002.50 2.50

Since s is multiplied by 2.50, divide both sides by 2.50. The symbol does not change.

Ryan can buy only a whole number of smoothies.

Ryan can buy 0, 1, 2, 3, 4, 5, or 6 smoothies.

CONFIDENTIAL 17

1) A pitcher holds 128 ounces of juice. What are the possible numbers of 10-ounce servings that one pitcher

can fill?

Now you try!

1) p ≤ 12.8

CONFIDENTIAL 18

Assessment


1) 10 < 2t

3) -80 < 8c 4) 21 > 3d

2) j ≤ 413

6) h ≤ 2 4 7

5) ≥ -2w4

1) t > 5 2) j ≤ 12

3) c > -10 4) d > 7

5) w ≥ -86) h ≤ 8 7

CONFIDENTIAL 19

Write an inequality for each sentence. Graph each inequality.

7) The product of a number and 7 is not less than 21.

8) The quotient of h and -6 is at least 5.

7) 7x ≥ 21

8) h ≥ 5 -6

CONFIDENTIAL 20

9) The rope Rosa brought with her camping gear is 54 inches long. Rosa needs to cut shorter pieces

of rope that are each 18 inches long. What are the possible number of pieces Rosa can cut?

10) What is the greatest possible integer solution of the inequality 3.806x < 19.902?

9) x ≤ 3

10) x ≤ 5

CONFIDENTIAL 21

Inequalities

Solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division,

undo the operation by dividing or multiplying both sides of the inequality by the same number.

STEP1: Identify the variable.

STEP2: To get the variable by itself, Multiply the same number to or Divide the same number from each side of the inequality .

STEP3: Check the solution .

Solving inequality by Multiply or Divide needs certain steps to be followed.

Let’s review

CONFIDENTIAL 22



When you multiply each side of a true inequality by a positive integer, the result remains true.


1. If a > b, then a × c > b × c and

2. If 7 > 2, then 7 × 4 > 2 × 4

Multiplication and Division by Positive Numbers

CONFIDENTIAL 23



When you divide each side of a true inequality by a positive integer, the result remains true.


1. If a > b, then a > b, c c

2. If 7 > 3, then 7 > 3, 4 4

CONFIDENTIAL 24

If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality

symbol to make the statement true.

5 > -3 5 is greater than -3.

Multiply both sides by -2.

You know that -10 is less than 6, so use the symbol for less than.

5 (-2) -3 (-2)

-10 6

-5 < 3

CONFIDENTIAL 25

Multiplying both sides by a negative number changes the sign of both sides of the inequality.

This means there is another set of properties of inequality for multiplying or dividing by a negative number.

0 2 4 6 8 1012 10 8 6 4 2

5

12 14

3

x(-2)

x(-2)

CONFIDENTIAL 26




1. If a > b, then a x c < b x c and

2. If 7 > 3, then 7 x -5 < 3 x -5



CONFIDENTIAL 27




1. If a > b, then a < b and c c

2. If 7 > 2, then 7 < 2, -4 -4



CONFIDENTIAL 28

A) -8x > 72

-8x > 72

x < -9

Since x is multiplied by -8, divide both sides by -8. Change > to <.

0 1 2 3 4 5 6 7 8 9 10 11 12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

x < -9

Multiplying or Dividing by a Negative Number


-8x < 72 -8 -8

CONFIDENTIAL 29

Problem Solving Application

Ryan has a $16 gift card for a health store where a smoothie costs $2.50 with tax. What are the possible

numbers of smoothies that Ryan can buy?

Let s represent the number of smoothies Ryan can buy.

$2.50 times number of smoothies is at most $16.00.

2.50 • s ≤ 16.00

s ≤ 6.4

2.50 • s ≤ 16.002.50 2.50

Since s is multiplied by 2.50, divide both sides by 2.50. The symbol does not change.

Ryan can buy only a whole number of smoothies.

Ryan can buy 0, 1, 2, 3, 4, 5, or 6 smoothies.

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CONFIDENTIAL 1 Algebra1 Solving Inequalities by Multiplying or Dividing