Question

Solve the inequality, graph the solution on the number line, and write the solution in interval notation:

$-5\le 2x-1<7$

Short answer

The solution of the inequality is$-2\le \hspace{0.17em}x<4$

Interval notation:$[-2,\hspace{0.17em}4)$

The graph of the inequality on number line is:

Step-by-step solution

Step 1. Given Information.

We have:

$-5\le 2x-1<\hspace{0.17em}7$

Step 2. Solve the compound inequality.

The inequality $-5\le 2x-1<\hspace{0.17em}7$can be written as $-5\le 2x-1$and$2x-1<\hspace{0.17em}7$ by using the property of inequality.

First, solve$-5\le 2x-1$

Switch sides:

$$\begin{gathered} 2x-1\ge \hspace{0.17em}-5 \\ 2x-1+1\ge \hspace{0.17em}-5+1 \\ 2x\ge \hspace{0.17em}-4 \\ \frac{2x}{2}\ge \frac{-4}{2} \\ x\ge \hspace{0.17em}-2 \end{gathered}$$

Interval notation:$[-2,\hspace{0.17em}\infty \hspace{0.17em})$

Now, solve $2x-1<\hspace{0.17em}7$:

$$\begin{gathered} 2x-1+1<7+1 \\ 2x<8 \\ \frac{2x}{2}<\frac{8}{2} \\ x<4 \end{gathered}$$

Interval notation:$\left(-\infty \hspace{0.17em},\hspace{0.17em}4\right)$

Step 3. Draw the solution on the number line.  

We have the solution $x\le -2$and $x>4$.

By combining the both solutions, we get $-2\le x<4$.

Draw the solution on the number line: