Chapter 5.5, Problem 31PPS

To determine

To solve the inequality:

  $7+t\le 2\left(t+3\right)+2$

And to graph the solution on a number line.

Answer to Problem 31PPS

  $\left[-1,\infty \right)$

  

Explanation of Solution

Given:

  $7+t\le 2\left(t+3\right)+2$

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

  $7+t\le 2\left(t+3\right)+2$

First, simplify this inequality, as

  $\begin{array}{c}7+t\le 2\left(t+3\right)+2\end{array}$

   $\begin{array}{c}\Rightarrow 7+t\le 2t+6+2\end{array}$

   $\begin{array}{c}\Rightarrow 7+t\le 2t+8\end{array}$

Now to simplify further, take like terms in one side of the $\le$ sign using the properties of inequality, as

  $\begin{array}{c}7+t\le 2t+8\end{array}$

   $\begin{array}{c}\Rightarrow 7+t-t\le 2t+8-t\end{array}$

   $\begin{array}{c}\Rightarrow 7\le t+8\end{array}$

   $\begin{array}{c}\Rightarrow 7-8\le t+8-8\end{array}$

   $\begin{array}{c}\Rightarrow -1\le t\end{array}$

Thus, the solution of given inequality is every number greater than and equals to $-1$ , this can be represented in interval form, as

  $\left[-1,\infty \right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point $-1$ is not strict, so use a solid circle at $-1$ . And the solution interval has $\infty$ at right endpoint which means every number greater than and equals to $-1$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: