Chapter 1.5, Problem 35PPSE

Solution

Whether the inequality $9x+2\left(2+x\right)<5+9x$ is always, sometimes, or never true?

Sometimes True

Given:

The inequality, $9x+2\left(2+x\right)<5+9x$ .

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.

Calculation:

In order to solve the given inequality $9x+2\left(2+x\right)<5+9x$ , perform the basic algebraic operations to isolate the variable term as shown below

  $\begin{array}{c}9x+2\left(2+x\right)<5+9x\\ 9x+4+2x<5+9x\\ 11x+4<5+9x\\ 11x-9x+4<5+9x-9x\\ 2x+4<5\\ 2x+4-4<5-4\\ 2x<1\\ x<\frac{1}{2}\end{array}$

Since the result is true for the values of x that are less than $\frac{1}{2}$ only, the given inequality is sometimes true.