### Linear Inequality

All the rules governing the solution of linear equations apply to the solution of linear inequalities, except for the differences listed below:

1.  Inequalities do not have equal signs. They are represented by the signs:  < which means, less than; which means, less than or equal to; > which means, greater than; which means, greater than or equal to.

2.  When an inequality is multiplied or divided by a negative number the sign changes. That is, a < (less than) sign would change to a > (greater than) sign, a (greater than or equal to) sign would change to a (less than or equal to) sign etc.

3.  The solution of an inequality is a range, which can be drawn on a number line, and is therefore written as a solution set.

Examples

Solve the following linear inequalities:

(a)     x + 7 ≥ 9

(b)    x + 2 < 10 – x

Solutions:

(a)

x + 7 ≥ 9

x + 7 – 7 ≥ 9 – 7

x ≥ 2

That is, {x: x ≥ 2}

(b)

x + 2 < 10 – x

x + x + 2 < 10 – x + x

2x + 2 -2 < 10 – 2

2x < 8

x < 4

That is, {x: x < 4}

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