SOLVING INEQUALITIES WITH VARIABLES ON BOTH SIDES

Some inequalities have variable terms on both sides of the inequality symbol. You can solve these inequalities like you solved equations with variables on both sides.

Use the properties of inequality to “collect” all the variable terms on one side and all the constant  terms on the other side.

Solving Inequalities with Variables on Both Sides

Solve each inequality and graph the solutions : 

Example 1 : 

4x + 10 ≤ 3x + 95

Solution :

4x + 10 ≤ 3x + 95

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

Example 2 : 

3m < 5m + 8

Solution :

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

Example 3 : 

5k - 1 ≤ 2.5k + 4

Solution :

5k - 1 ≤ 2.5k + 4

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

Simplifying Each Side Before Solving

Solve each inequality and graph the solutions.

Example 4 : 

6(1 - m) < 3m

Solution :

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

Example 5 : 

1.5k < -0.3(k + 1) + 1.2

Solution :

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

All Real Numbers as Solutions or No Solutions

Example 6 : 

y + 4 ≥ y + 3

Solution :

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.

Example 7 : 

2(p + 3) < 5 + 2p

Solution :

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.

Business Application

Example 8 : 

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?

Solution : 

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.

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