Equations & Inequalities: The Final Challenge

Equations

Ex. 1) 3(2x + 5) = -9 1. Distribution

6x + 15 = -9– 15 – 15

6x = -24 6 6

x = -4

2. Subtract 15 from both sides.

3. Divide both sides by 6.

Ex. 2) 5

8

3

2x

The goal is to get 1x alone.

To do this, multiply

both sides by ,

which is the reciprocal

of .

2

3

3

2

2

3

5

8

3

2

2

3x

Reduce fractions if possible.

4

1

Multiply fractions, and simplify.5

22

5

121 x

Ex. 3) 22

15 x Subtract 5 from both sides

to get the x term alone.

5 5

1

2-

1

2-

141

14x

1

7

2

1

x Multiply both sides by ,

which is the reciprocal of .

1

2

2

1

Reduce fractions if possible.

Multiply fractions, and simplify.

Ex. 4) 15465 x

Add 4 to both sides to get 1x alone.4 4

To eliminate the fraction,

multiply both sides by ,

which is the reciprocal of .

5

6

6

5

56

115

65

56 4x

Reduce fractions if possible.

-3

1

184 x

14x

Ex. 5) The perimeter of Mr. Mac Gregor’s garden is 64 meters. The length of the garden in 20 meters. What is the width of the garden? Write and solve an equation to answer this question!

Let w = width of the rectangle

The perimeter is the distance around the outside of the rectangle.

P = 2(Length) + 2(width)P = 2L + 2w

- 40 -4024 = 2w

2 2

12 m = w The width of the garden is 12 meters.

20 m

20 m

w w

64 = 2(20) + 2w64 = 40 + 2w

Ex. 6) Students at the rec center are taking a trip to the county fair. The cost of the trip is $52 per student. This price includes a concert ticket worth $11 and 2 passes for the rides and the game booths. If the passes are worth the same amount of money, how much does 1 pass cost? Write and solve an equation to answer this question!

Let x = cost of 1 pass

Total cost = cost of concert ticket + cost of 2 passes

- 11 -11

41 = 2x2 2

$20.50 = x One pass would cost $20.50.

52 = 11 + 2x

Ex. 7) 15 ≥ -2x + 5 -5 -5 10 ≥ -2x -2 -2 -5 ≤ x

x ≥ -5

Subtract 5 from both sides to get the x term alone.

Divide both sides by -2.

-7 -6 -5 -4 -3

Graph your answer on a number line.

Reverse the direction of the inequality symbol because you divided by a negative number.

Inequalities

Let x = the amount of miles driven

45 + 0.20x < 100 -45 -45 0.20x < 55 0.20 0.20 x < 275

Ex. 8) A rental car company charges $45 plus $0.20 per mile to rent a car. Cindy wants to spend less than $100 for her car rental. How many miles she can drive and keep the bill under $100?

Cindy must drive less than 275 miles.