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Inequalities Solutions

An inequality represents two expressions that are not equal to each other.
The symbols that represent inequalities are:

Symbol	In words	On a number line
≤	is less than or equal to	⏺
≥	is greater than or equal to	⏺
<	is less than	○
>	is greater than	○

A solution to an inequality is infinite because it represents a range of values.
Use an open or closed point to show that a value is or is not included as a solution.
Use shading on a number line to represent all values that are true for the inequality.

Example 1

Solve. Graph the solutions on a number line.

$2 x - 7 < - 5$

Implement	Explain
$\begin{array}{rcl} 2 x - 7 & < & - 5 \\ + 7 + 7 \end{array}$	Add 7 to both sides
$\begin{array}{rcl} (\frac{1}{2}) 2 x & < & 2 (\frac{1}{2}) \end{array}$	Multiply by $\frac{1}{2}$ on both sides
$\begin{array}{rcl} x & < & 1 \end{array}$	Open point, shade to the left of 1

Example 2

Solve. Graph the solutions on a number line.

$\frac{2}{3} (x - 3) > 4$

Implement	Explain
$\begin{array}{rcl} \frac{2}{3} x - 2 & > & 4 \\ + 2 + 2 \\ (\frac{3}{2}) \frac{2}{3} x & > & 6 (\frac{3}{2}) \\ x & > & 9 \end{array}$	Distribute Add 2 to both sides Multiply by $\frac{3}{2}$ on both sides Open point, shade to the right of 9