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Unit 3 Day 7: Solving Inequalities with Variables on Both Sides

- Essential Questions: How do we solve inequalities with variables on both sides? When does an inequality have no solution or a solution of all real numbers?

Vocabulary

- No solution: when the variable in an equation or inequality is eliminated and you are left with a false statement
- All real numbers: when the variable in an equation or inequality is eliminated and you are left with a true statement

- x

- < 1

- > 4

- 9

- 9

- 9

- 9

- 9x

- 9x

- > 36

- < 9

- Example 1: Solve the inequalities.
- 7x + 19 > -2x + 55 6x + 22 < -3x + 31

- 9x

- 9x

- + 19

- + 22

- > 55

- < 31

- + 2x

- + 2x

- + 3x

- + 3x

- - 19

- -19

- - 22

- -22

- 1

- -2

- 2

- -12

- -12

- Example 2: Solve the inequalities.
- x + 2 > 3x + 1 -8x + 7 < 4x – 5

- -12x

- < -12

- -2x

- > -1

- x <

- -2x + 2

- > 1

- -12x + 7

- < - 5

- - 3x

- - 3x

- - 4x

- - 4x

- - 7

- - 7

- - 2

- - 2

- x > 1

- + 4

- -6

- -6

- 4

- -6x

- < 12

- 1

- -6x + 4

- < 16

- Example 3: Solve the inequality.
- (-12x + 16) < 10 – 3(-x – 2)

- -3x

- + 4

- + 3x

- + 6

- < 10

- < 16

- + 3x

- - 3x

- - 3x

- - 4

- - 4

- x > -2

- < 16

- 2

- 16

- 16

- Example 4: Solve the inequality.
- (12x – 4) < 2(7 – 5x)

- 1

- 16x

- - 2

- < 14

- 6x

- - 2

- < 14

- - 10x

- + 10x

- + 10x

- + 2

- + 2

- x < 1

- 3

- - x

- + 3

- > 3

- Example 5: Solve the inequalities.
- 12 – 2a < - 5a – 9 x – 2x + 3 > 3 – x

- 3a

- < - 21

- > 3

- 3

- 12

- + 3a

- < - 9

- + 5a

- + 5a

- - x

- + x

- + x

- - 12

- - 12

- true statement
- infinite solutions

- a < -7

- - 4

- + 6

- -2y

- 5x

- > -10

- < -25

- - 4

- + 24

- > 5y

- 24

- -2

- -2

- Example 6: Solve the inequalities.
- 5x + 24 < 5(x - 5) 6y - (3y - 6) > 5y - 4

- -2y + 6

- >

- - 4

- < 5x

- - 25

- + 6

- 6y

- - 3y

- - 5x

- - 5x

- 3y

- - 5y

- - 5y

- false statement
- no solutions

- - 6

- - 6

- y < 5

- .03

- .36 > .03x

- Example 7: Phone Company A charges an activation fee of 36 cents and then 3 cents per minute. Phone Company B charges 6 cents per minute with no activation fee. For what value of x is Phone Company A more expensive than Phone Company B?

- .36 + .03x > .06x

- Phone Company A is more expensive when the number of minutes is less than 12. If you talk for more than 12 minutes, Phone Company A is a good choice.

- - .03x

- - .03x

- 12 > x

- x < 12

- 3

- 150 + 3x < 195

- Example 8: Justin and Tyson are beginning an exercise program to train for football season. Justin weighs 150 pounds and hopes to gain 2 pounds per week. Tyson weighs 195 pounds and hopes to lose 1 pound per week. If the plan works, for how many weeks will Justin weigh less than Tyson?

- 3x < 45

- Justin

- Tyson

- < 195 - 1x

- 150 + 2x

- + 1x

- + 1x

- Justin will weigh less than Tyson up until the 15 week mark.

- - 150

- - 150

- x < 15

Essential Questions:How do we solve inequalities with variables on both sides? When does an inequality have no solution or a solution of all real numbers?

- Take 1 minute to write 2 sentences answering the essential question.

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