Chapter 5.2, Problem 2CYU

To determine

To find the value of $n$ that satisfied the inequality $30>\frac{1}{2}n$ and show on the number line.

Answer to Problem 2CYU

The value of $n$ that satisfied the inequality $30>\frac{1}{2}n$ is $n<60$ and the number line is given as:

  

Explanation of Solution

Given:

The inequality is given as $30>\frac{1}{2}n$ .

Concept Used:

• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.endpoint.
• To graph endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Properties of inequality:

 1. $b\le c\Rightarrow ab\le ac,\text{ if }a\ge 0$

 2. $b\le c\Rightarrow ab\ge ac,\text{ if }a\le 0$

Calculation:

Simplify the inequality $30>\frac{1}{2}n$ for the value of $n$ .

First, switch the sides of the inequality:

  $\frac{1}{2}n<30$

Multiply both sides by 2:

  $\begin{array}{c}2\cdot \frac{1}{2}n<30\cdot 2\\ \cancel{2}\cdot \frac{1}{\cancel{2}}n<60\\ n<60\end{array}$

Thus, the inequality is true for m less than $60$ , thus the solution of given inequality in interval form is $\left(-\infty ,60\right)$

Hence, the value of $n$ that satisfied the inequality $30>\frac{1}{2}n$ is $n<60$ .

Now in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point $60$ is strict, so use a blank circle at $60$ . And the solution interval has $-\infty$ at left endpoint which means every number less than $60$ to negative infinity is in the solution interval and does not include the right end point as it is an open interval. The graph of the inequality is shown below: