SOLVING INEQUALITIES WITH VARIABLES ON BOTH SIDES WORKSHEET

Problems 1-8 : Solve each inequality and graph the solutions.

Problem 1 : 

3m < 5m + 8

Problem 2 : 

4x + 10 ≤ 3x + 95

Problem 3 : 

-2a - 6 > 5a + 1

Problem 4 : 

5k - 1 ≤ 2.5k + 4

Problem 5 : 

y - 43 ≤ 4y - 22

Problem 6 : 

(2 - 3z)/2 < 3 - 2.5z

Problem 7 : 

6(1 - m) < 3m

Problem 8 : 

1.5k < -0.3(k + 1) + 1.2

Problems 9-10 : Solve each inequality and determine if all real numbers are solutions or no solutions.

Problem 9 : 

y + 4 ≥ y + 3

Problem 10 : 

2(p + 3) < 5 + 2p

Problem 11 : 

The Daily Info charges a fee of $675 plus $75 per week to run an ad. The People’s Paper charges $150 per week. For how many weeks will the total cost at Daily Info be less expensive than the cost at People’s Paper?

Problem 12 : 

A-Plus Advertising charges a fee of $24 plus $0.10 per flyer to print and deliver flyers. Print and More charges $0.25 per flyer. Mr. Johnson wants to print and deliver 180 flyers. Determine the preferred course of action.

Detailed Answer Key

1. Answer :

3m < 5m + 8

To collect the variable terms on one side, subtract 2m from each side.

0 < 2m + 8

Because 8 is added to 2m, subtract 8 from each side to undo the addition.

-8 < 2m

Because m is multiplied by 2, divide each side by 2 to undo the multiplication.

-4 < m

m > -4

2. Answer :

4x + 10 ≤ 3x + 95

Subtract 3x from each side. 

x + 10 ≤ 95

Because 10 is added to x, subtract 100 from each side to undo the addition.

x ≤ 85

3. Answer :

-2a - 6 > 5a + 1

Add 2a to each side. 

-6 > 7a + 1

Because 1 is added to 7a, subtract 1 from each side to undo the addition.

-7 > 7a

Because a is multiplied by 7, divide each side by 7 to undo the multiplication. 

-1 > a

a < -1

4. Answer :

5k - 1 ≤ 2.5k + 4

Subtract 2.5k from each side.

2.5k - 1 ≤ 4

Because 1 is subtracted from 2.5k, add 1 to each side to undo the subtraction.

2.5k ≤ 5

Because k is multiplied by 2.5, divide each side by 2.5 to undo the multiplication.

k ≤ 2

5. Answer :

y - 43 ≤ 4y - 22

Subtract y from each side. 

-43 ≤ 3y - 22

Because 22 is subtracted from 3y, add 22 to each side to undo the subtraction.

-21 ≤ 3y

Because y is multiplied by 3, divide each side by 3 to undo the multiplication. 

-7 ≤ y

y ≥ -7

6. Answer :

(2 - 3z)/2 < 3 - 2.5z

Because (2 - 3z) is divided by 2, multiply each side by 2 to undo the division.

2[(2 - 3z)/2] < 2(3 - 2.5z)

2 - 3z < 6 - 5z

Add 5z to each side. 

2 + 2z < 6

Because 2 is added to 2z, subtract 2 from each side to undo the addition.

2z < 4

Because z is multiplied by 2, divide each side by 2 to undo the multiplication.

z < 2

7. Answer :

6(1 - m) < 3m

Distribute 6 on the left side of the inequality.

6 - 6m < 3m

Add 6m to each side so that the coefficient of m is positive.

6 < 9m

Because m is multiplied by 9, divide each side by 9 to undo the multiplication.

2/3 < m

m > 2/3

8. Answer :

1.5k < -0.3(k + 1) + 1.2

Distribute -0.3 on the right side of the inequality.

1.5k < -0.3k - 0.3 + 1.2

Combine the like terms on the right side. 

1.5k < -0.3k + 0.9

Add 0.3k to each side. 

1.8k < 0.9

Because k is multiplied by 1.8, divide each side by 1.8 to undo the multiplication.

k < 1/2

9. Answer :

y + 4 ≥ y + 3

The same variable term (y) appears on both sides. Look at the other terms.

For any number y, adding 4 will always result in a greater number than adding 3.

All values of x make the inequality true.

All real numbers are solutions.

10. Answer :

2(p + 3) < 5 + 2p

Distribute 2 on the left side.

2p + 6 < 5 + 2p

The same variable term (2p) appears on both sides. Look at the other terms.

For any number 2p, adding 6 will never result in a lesser number than adding 5.

No values of x make the inequality true.

There are no solutions.

11. Answer : 

Let x be the number of weeks the ad runs in the paper.

675 + 75x < 150x

Subtract 75x from each side. 

675 < 75x

Because x is multiplied by 75, divide each side by 75 to undo the multiplication.

9 < x

x > 9

The total cost at Daily Info is less than the cost at People’s Paper, if the ad runs for more than 9 weeks.

12. Answer : 

Let y be the number of flyers.

24 + 0.10y < 0.25y

Subtract 0.10y from each side. 

24 < 0.15y

Because y is multiplied by 0.15, divide each side by 0.15 to undo the multiplication.

160 < y

y > 160

The cost at A-Plus Advertising less than the cost at Print and More for more than 160 flyers.

Because Mr. Johnson wants to print and deliver 180 (more than 160) flyers, the cost at A-Plus Advertising is less than the cost at Print and More. 

So, A-Plus Advertising is preferable. 

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