Solving Linear Inequalities

 
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The rules for solving inequalities are similar to those for solving linear equations. However, there is one exception when multiplying or dividing by a negative number.

To solve an inequality, we can:

• Add the same number to both sides.
• Subtract the same number from both sides.
• Multiply both sides by the same positive number.
• Divide both sides by the same positive number.
• Multiply both sides by the same negative number and reverse the sign.
• Divide both sides by the same negative number and reverse the sign.

Inequalities Of The Form “x + a > b” or “x + a < b”

Example:

Solve x + 7 < 15

Solution:

x + 7 < 15
x + 7 – 7 < 15 – 7
x < 8

Inequalities Of The Form “x – a < b” or “x – a > b”

Example :

Solve x – 6 > 14

Solution:

x – 6 > 14
x – 6+ 6 > 14 + 6
x > 20

Example :

Solve the inequality x – 3 + 2 < 10

Solution:

x – 3 + 2 < 10
x – 1 < 10
x – 1 + 1 < 10 + 1
x < 11

Inequalities Of The Form “a – x < b” or “a – x > b”

Example :

Solve the inequality 7 – x < 9

Solution:

7 – x < 9
7 – x – 7 < 9 – 7
– x < 2
x > –2 (remember to reverse the symbol when multiplying by –1)

Example :

Solve the inequality 12 > 18 – y

Solution:

12 > 18 – y
18 – y < 12
18 – y – 18 < 12 –18
– y < –6
y > 6 (remember to reverse the symbol when multiplying by –1)

Inequalities Of The Form “ < b” or “ > b”

Example:

Solve > 3

Solution:

> 3

× 5 > 3 × 5

x > 15

Example:

Solve

Solution:

Solving Linear Inequalities With Like Terms

If an equation has like terms, we simplify the equation and then solve it. We do the same when solving inequalities with like terms.

Example :

Evaluate 3x – 8 + 2x < 12

Solution:

3x – 8 + 2x < 12
3x + 2x < 12 + 8
5x < 20
x < 4

Example:

Evaluate 6x – 8 > x + 7

Solution:

6x – 8 > x + 7
6x – x > 7 + 8
5x > 15
x > 3

Example:

Evaluate 2(8 – p) ≤ 3(p + 7)

Solution:

2(8 – p) ≤ 3(p + 7)
16 – 2p ≤ 3p + 21
16 – 21 ≤ 3p + 2p
–5 ≤ 5p
–1 ≤ p
p ≥ –1 (a < b is equivalent to b > a)

Videos

An introduction to solving inequalities

An Introduction To Solving Linear Inequalities