Solving Inequalities

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

Solving Linear Inequalities -
Examples are shown of how to solve linear inequalities

Solving inequalities -
Students learn that when solving an inequality, such as --3x is less than 12, the goal is the same as when solving an equation: to get the variable by itself on one side.
Note that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be switched.
For example, to solve --3x is less than 12, divide both sides by --3, to get x is greater than -4.
And when graphing an inequality on a number line, less than or greater than means an open dot, and less than or equal to or greater than or equal to means a closed dot.

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