Chapter 1.6, Problem 35PPSE

Solution

To solve the inequality $3\left|2x-1\right|\ge 21$ and graph the solution.

  $x\le -3\text{or}x\ge 4$

The graph of solution is shown below:

  

Given:

The inequality, $3\left|2x-1\right|\ge 21$ .

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$ , use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$ , use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.
• The absolute value property: $\text{if }\left|x\right|>a,\text{then}x>a\text{ or }x<-a$

Calculation:

In order to solve the given inequality $3\left|2x-1\right|\ge 21$ , use the absolute value property and isolate the variable on one side by performing the basic algebraic operations as shown below

  $\begin{array}{c}3\left|2x-1\right|\ge 21\\ \frac{3\left|2x-1\right|}{3}\ge \frac{21}{3}\\ \left|2x-1\right|\ge 7\\ 2x-1\le -7\text{or}2x-1\ge 7\\ 2x-1+1\le -7+1\text{or}2x-1+1\ge 7+1\\ 2x\le -6\text{or}2x\ge 8\\ \frac{2x}{2}\le \frac{-6}{2}\text{or}\frac{2x}{2}\ge \frac{8}{2}\\ x\le -3\text{or}x\ge 4\end{array}$

Thus, the solution of the inequality is $x\le -3\text{or}x\ge 4$ .

Since the inequality sign is not strict with both endpoints, use the solid circles at both endpoints and shade the region to the left of $-3$ and to the right side of 4 as shown below