Question

Explain the steps for solving the compound inequality:

$2-7x\ge 5$or

$4\left(x-3\right)+7>\hspace{0.17em}3$

Short answer

To solve the compound inequality first solve both inequalities separately and then the solution to the compound inequality will contain the solution of the both inequality.

Step-by-step solution

Step 1. Given information. 

We have:

$2-7x\ge 5$or

$4\left(x-3\right)+7>\hspace{0.17em}3$

Step 2. Solve the inequalities separately. 

First, solve $2-7x\ge 5:$

Subtract $2$from both sides:

role="math" localid="1644909590831" $2-7x-2\ge \hspace{0.17em}5-2$

$-7x\ge \hspace{0.17em}3$

Multiply both sides by $-1$(reverse the inequality):

$$\begin{gathered} \left(-7x\right)\left(-1\right)\le \hspace{0.17em}3\left(-1\right) \\ 7x\le \hspace{0.17em}-3 \\ \frac{7x}{7}\le \frac{-3}{7} \\ x\le \hspace{0.17em}-\frac{3}{7} \end{gathered}$$

Step 3. Solve the second inequality. 

We have:

$4\left(x-3\right)+7>\hspace{0.17em}3$

Subtract $7$from both sides:

$$\begin{gathered} 4\left(x-3\right)+7-7>3-7 \\ 4\left(x-3\right)>-4 \\ \frac{4\left(x-3\right)}{4}>\frac{-4}{4} \\ x-3>-1 \\ x-3+3>-1+3 \\ x>2 \end{gathered}$$

Step 4. Merge the overlapping intervals. 

The solution to the compound inequality will include both solutions.

Thus, the solution to the compound inequality is:

$x\le 1\cup x>2$