Chapter 1.6, Problem 5LC

Solution

To solve the inequality $\left|4x-6\right|\ge 10$ and graph the solution.

  $x\le -1\text{or}x\ge 4$

The graph of solution is shown below:

  

Given:

The inequality, $\left|4x-6\right|\ge 10$ .

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 $\left|4x-6\right|\ge 10$ , use the absolute value property and isolate the variable on one side by performing the basic algebraic operations as shown below

  $\begin{array}{c}\left|4x-6\right|\ge 10\\ 4x-6\ge 10\text{or}4x-6\le -10\\ 4x-6+6\ge 10+6\text{or}4x-6+6\le -10+6\\ 4x\ge 16\text{or}4x\le -4\\ \frac{4x}{4}\ge \frac{16}{4}\text{or}\frac{4x}{4}\le \frac{-4}{4}\\ x\ge 4\text{or}x\le -1\end{array}$

Thus, the solution of the inequality is $x\le -1\text{or}x\ge 4$ .

Since the inequality sign is not strict with both endpoints, use the solid circles at both endpoints and shade to the left of $-1$ and to the right of 4 as shown below