Chapter EP, Problem 1.5.17EP

To determine

To solve: The inequality $5\left(x-1\right)-4x\ge 3\left(3-x\right)$ and then plot the graph of the solution set on a number line.

Answer to Problem 1.5.17EP

The solution of the inequality $5\left(x-1\right)-4x\ge 3\left(3-x\right)$ is $x\ge \frac{7}{2}$ , and graph of the solution set of the inequality is shown in figure (1).

Explanation of Solution

Given:

The inequality is $5\left(x-1\right)-4x\ge 3\left(3-x\right)$ .

Calculation:

Simplify the inequality as follows.

  $\begin{array}{c}5\left(x-1\right)-4x\ge 3\left(3-x\right)\\ 5x-5-4x\ge 9-3x\\ x-5\ge 9-3x\\ 4x\ge 14\end{array}$

Divide both sides of the inequality $4x\ge 14$ by $4$ .

  $\begin{array}{c}\frac{4x}{4}\ge \frac{14}{4}\\ x\ge \frac{7}{2}\end{array}$

Therefore, the solution of the inequality $5\left(x-1\right)-4x\ge 3\left(3-x\right)$ is $x\ge \frac{7}{2}$ .

The graph of the solution set of the inequality is as follows.

Figure (1)

The graph for the inequality $x\ge \frac{7}{2}$ contains all values that are greater than or equal to $\frac{7}{2}$ and dot shows $\frac{7}{2}$ is included in solution set.

Therefore, the solution of the inequality $5\left(x-1\right)-4x\ge 3\left(3-x\right)$ is $x\ge \frac{7}{2}$ , and graph of the solution set of the inequality is shown in figure (1).