# Inequalities » Linear inequalities

Check what is an inequality if you do not know what an inequality is.
Of course it is important to know what all the signs mean.
Check domain and range if your forgot what <, >, ≤ and ≥ mean.

## Contents

1. Direction of sign afterwards
2. Direction of sign during solving

## 1. Direction of sign afterwards

Steps:

 1 Make an equation that corresponds to the inequality. 2 Solve this equation. 3 Fill in a number greater than and less than the solution en see which are correct with the inequality (you may use a number line to indicate these). 4 Write the solution of the inequality.

Example 1

 8a + 4 < 5a + 16 8a + 4 = 5a + 16 8a = 5a + 12 3a = 12 a = 4

Try a = 3, you get
 8 × 3 + 4 < 5 × 3 + 16 28 < 31 Cor rect!
Try a = 5, you get
 8 × 5 + 4 < 5 × 5 + 16 44 < 41 Inco rrect!

The inequality is correct for numbers less than 4.
Solution: a < 4

Example 2

 3w + 7 < 9w – 23 3w + 7 = 9w – 23 3w = 9w – 30 –6w = –30 w = 5

Try w = 4, you get
 3 × 4 + 7 < 9 × 4 – 23 19 < 13 Inco rrect!
Try w = 6, you get
 3 × 6 + 7 < 9 × 6 – 23 25 < 31 Cor rect!

The inequality is correct for numbers greater than 5.
Solution: w > 5

## 2. Direction of sign during solving

This method is faster, as it is less work. However, you make a mistake much more easily. You easily forget to apply the rule and you often do not check your answer at the end. This method can only be used when there is one solution for the equation.

The rule:
When multiplying or dividing with a negative number because of the balance method, the sign changes direction.

And:
When you swap your inequality, so you change the right-hand side of the inequality to be the left-hand side and vice versa, the sign also changes direction!

Examples

 7x + 5 < 9x + 4 –5 –5 7x < 9x – 1 –9x –9x –2x < –1 : –2 : –2 x > 0,5
 –4x – 20 > –8x – 8 +20 +20 –4x > –8x + 12 +8x +8x 4x > 12 :4 :4 x > 3

 5x + 3 < 9x + 23 –5x –5x 3 < 4x + 23 –23 –23 –20 < 4x swapping 4x > –20 :4 :4 x > –5

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