Solving Two Step Linear Inequalities

Example 1:

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.

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

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 $3$ .

Example 2:

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.

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

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 $-2$ .