Chapter 5.3, Problem 9CYU

To determine

To solve the inequality:

  $-6\le 3\left(5v-2\right)$

And to graph the solution on a number line.

Answer to Problem 9CYU

  $\left[0,\infty \right)$

  

Explanation of Solution

Given:

  $-6\le 3\left(5v-2\right)$

Concept Used:

• 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.
• Properties of inequality:

1. $b\le c\Rightarrow b\pm a\le c\pm a$
2. $b\le c\Rightarrow ab\le ac,\text{ if }a>0$
3. $b\le c\Rightarrow ab\ge ac,\text{ if }a<0$

Calculation:

In order to solve the inequality

  $-6\le 3\left(5v-2\right)$

First, simplify this inequality as,

  $\begin{array}{c}-6\le 3\left(5v-2\right)\\ \Rightarrow -6\le 15v-6\end{array}$

Now, add 6 both sides of the inequality and the simplify further using properties of the inequality, as

  $\begin{array}{c}-6\le 15v-6\\ \Rightarrow -6+6\le 15v-6+6\\ \Rightarrow 0\le 15v+0\\ \Rightarrow \frac{0}{15}\le \frac{15v}{15}\\ \Rightarrow 0\le v\end{array}$

Thus, the solution of given inequality is every number greater than and equals to 0, this can be represented in interval form, as

  $\left[0,\infty \right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point 0 is not strict, so use a solid circle at 0. And the solution interval has $\infty$ at right endpoint which means every number greater than and equals 0 is in the solution interval of given inequality. The graph of the solution of inequality is shown below: