Question

Solve each inequality. Describe the solution set using set-builder or interval notation. Then graph the solution set on number line.

$\frac{5z+2}{4}<\frac{5z}{4}+2$

Short answer

Solution of inequality $\frac{5z+2}{4}<\frac{5z}{4}+2$ is all real number

In interval form, solution set is $\left(-\infty ,\infty \right)$

Solution set on number line is

Step-by-step solution

- Write properties of inequality

Properties of inequalities are written in following table:

- Simplify 5z+24<5z4+2

To simplify given inequality, use subtraction and multiplication property of inequalities. Subtract by same number does not change the sign of inequality. Also multiply by same positive number does not change the sign of inequality as shown below

$\begin{array}{c}\frac{5z+2}{4}<\frac{5z}{4}+2\\ \frac{5z+2}{4}\times 4<\left(\frac{5z}{4}+2\right)\times 4\\ 5z+2<5z+8\\ 5z+2-5z<5z+8-5z\\ 2<8\end{array}$

- Write solution set

As $2<8$ is always true, so given inequality is true for all real numbers

The solution set of the inequality contains all real numbers

In interval form solution set is $\left(-\infty ,\infty \right)$

- Graph the solution set

Assolution set of the inequality is all real numbers

So arrow is towards both $-\infty$and $\infty$