Chapter P, Problem 60RE

Solution

The solution of the given inequality, and plot the solution on number line.

  $\left[-\frac{5}{3},\infty \right)$ ,

  

Given information:

The inequality given is:

  $5x+1\ge 2x-4$ .

Formula 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 given inequality:

  $5x+1\ge 2x-4$

Take all the terms of variable on one side of the inequality and rest on other side of the inequality, as

  $\begin{array}{c}5x-2x\ge -4-1\\ 3x\ge -5\end{array}$

Now, divide both sides by 3, as:

  $\begin{array}{c}\frac{3x}{3}\ge -\frac{5}{3}\\ x\ge -\frac{5}{3}\end{array}$

Thus, the solution of given inequality is every number greater than and equals to $-\frac{5}{3}$ , this can be represented in interval form, as

  $\left[-\frac{5}{3},\infty \right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point, $-\frac{5}{3}$ , is not strict, so use a solid circle at $-\frac{5}{3}$ . And the solution interval at right endpoint is $\infty$ , i.e. the solution is every number bigger than $-\frac{5}{3}$. The graph of the solution of inequality is shown below: