Question

Solve each inequality. Graph the solution set on a number line.

$\left|7x\right|+4<0$

Short answer

Given inequality has no solution

The solution set in the number line is empty

Step-by-step solution

- Definition of absolute value inequality

For real numbers $a\text{and}b$ with $b>0$

Absolute value inequality is $\left|a\right|<b$ when $-b<a\text{and}a<b$, and

Absolute value inequality is $\left|a\right|>b$ when $a<-b\text{or}a>b$

- Write properties of inequality

Properties of inequalities are written in the following table:

- Simplify 7x+4<0

Use subtraction property of inequality, subtracting with the same number does not change the sign of inequality as shown below

$\begin{array}{c}\left|7x\right|+4-4<0-4\\ \left|7x\right|<-4\end{array}$

From the definition of absolute value, it is the distance of a point from 0, so it is always greater than or equal to 0. But in the given expression, the absolute value is less than 0 so it has no solution.

- Graph the solution set

As given expression has no solution set, so it cannot be represented on the number line.

Thus, there is no solution on the number line, so it is empty.