Chapter 5, Problem 2MCQ

To determine

To solve the inequality:

  $m+2\ge 6$

And to graph the solution on a number line.

Answer to Problem 2MCQ

  $\left[4,\infty \right)$

  

Explanation of Solution

Given:

  $m+2\ge 6$

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

  $m+2\ge 6$

Subtract 2 from both sides of the inequality, as

  $\begin{array}{c}m+2\ge 6\\ \Rightarrow m+2-2\ge 6-2\\ \Rightarrow m+0\ge 4\\ \Rightarrow m\ge 4\end{array}$

Thus, the solution of given inequality is every number greater than and equals to 4, this can be represented in interval form, as

  $\left[4,\infty \right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point 4 is not strict, so use a solid circle at point 4. And the solution interval at right endpoint $\infty$ , thus every point greater than 4 is in the solution interval of given inequality. The graph of the solution of inequality is shown below: