Question

Solve the inequality. Graph the solution set.

$\left|4x+1\right|\ge 7$

Short answer

The solution set of the given inequality is:

$(-\infty ,-2]\cup [\frac{3}{2},\infty )$

The graph of the solution set is given as:

Step-by-step solution

Step 1. Given information 

The given inequality is$\left|4x+1\right|\ge 7$

Step 2.  Open the modulus function 

We know that, from the principle of absolute value, for $\left|x\right|\ge a$, we have:

$-a\ge x\ge a$.

Similarly, for $\left|4x+1\right|\ge 7$, we have:

$-7\ge 4x+1\ge 7$

Step 3. Solving the inequality 

Solving the inequality using properties, we get :

$$\begin{gathered} -7\ge 4x+1\ge 7 \\ -7-1\ge 4x\ge 7-1 \\ -8\ge 4x\ge 6 \\ -2\ge x\ge \frac{3}{2} \end{gathered}$$

Step 4. Verify the solution set 

Substituting $x=2$in the given inequality, we get:

$$\begin{gathered} \left|4x+1\right|\ge 7 \\ \left|4\times \left(-2\right)+1\right|\ge 7 \\ \left|-8+1\right|\ge 7 \\ \left|-7\right|\ge 7 \\ 7\ge 7 \end{gathered}$$

This is true.

Hence, the solution set of the given inequality is$-2\ge x\ge \frac{3}{2}$or$x\le -2orx\ge \frac{3}{2}$

Step 5. Graph the solution set 

The graph of the solution set is given as: