# Solving Two Step Linear Inequalities

To solve a two-step inequality, undo the addition or subtraction first, using inverse operations , and then undo the multiplication or division.

The inverse operation of addition is subtraction and vice versa.

Similarly, the inverse operation of multiplication is division and vice versa.

Note that, whenever you multiply or divide both sides of an inequality by a negative number, reverse the inequality.

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Example 1:
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Solve $2x+1<7$ .

First, we need to isolate the variable term on one side of the inequality. Here, on the left, $1$ is added to the variable term, $2x$ . The inverse operation of addition is subtraction. So, subtract $1$ from both sides.

$\begin{array}{l}2x+1-1<7-1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2x<6\end{array}$

Now, we have the variable $x$ multiplied by $2$ . The inverse operation of multiplication is division. So, divide both sides by $2$ .

$\begin{array}{l}\frac{2x}{2}<\frac{6}{2}\\ x<3\end{array}$

That is, the inequality is true for all values of $x$ which are less than $3$ .

Therefore, the solutions to the inequality $2x+1<7$ are all numbers less than $3$ .
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Example 2:
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Solve $-3x-8\ge -2$ .

First we need to isolate the variable term $-3x$ on the left. The inverse operation of subtraction is addition. So, add $8$ to both sides.

$\begin{array}{l}-3x-8+8\ge -2+8\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-3x\ge 6\end{array}$

To isolate the variable $x$ , divide both sides by $-3$ .

Note that, whenever you multiply or divide both sides of an inequality by a negative number, reverse the inequality.

$\begin{array}{l}\frac{-3x}{-\mathrm{3\hspace{0.17em}}}\le \frac{\mathrm{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}6}}{-3}\\ x\le -2\end{array}$

Therefore, the solutions to the inequality $-3x-8\ge -2$ are all numbers less than or equal to $-2$ .