Properties of Inequalities and Solving Linear Inequalities

Properties of Inequalities

(i) Addition and subtraction properties of inequalities. For any real number $a,b$ and $c$ : If $a > b$ , then $a + c > b + c$ and $a - c > b - c$ If $a < b$ , then $a + c < b + c$ and $a - c < b - c$

(ii) Multiplication and division properties of inequalities. If $c$ is positive and $a < b$ , then $a c < b c$ and $\frac{a}{c}<\frac{b}{c}$$\frac{a}{c} < \frac{b}{c}$ If $c$ is positive and $a > b$ , then $a c > b c$ and $\frac{a}{c}>\frac{b}{c}$$\frac{a}{c} > \frac{b}{c}$ If $c$ is negative and $a < b$ , then $a c > b c$ and $\frac{a}{c}>\frac{b}{c}$$\frac{a}{c} > \frac{b}{c}$ If $c$ is negative and $a > b$ , then $a c < b c$ and $\frac{a}{c}<\frac{b}{c}$$\frac{a}{c} < \frac{b}{c}$

Example: Solve $6x - 7\geq 2x + 17$. Graph the solution.

Solution: $6x - 7\geq 2x + 17$

$6x - 7 + 7\geq 2x + 17 + 7$ (adding 7 on both sides)

$6x - 2x\geq 2x - 2x + 24$ (subtracting $2x$ from both sides)

$4x\geq 24$ (dividing both sides by 4)

$x\geq 6$

Example: Solve and graph the solution

$10.7x - 6 + 5.5x\geq 3(5x - 3)$