Q38. - Algebra 2

Question

Solve each inequality. Describe the solution set using set-builder or interval notation. Then graph the solution set on number line. 

$12\left(\frac{1}{4}-\frac{n}{3}\right)\le -6n$

Step-by-Step Solution

Answer

Solution of inequality $12\left(\frac{1}{4}-\frac{n}{3}\right)\le -6n$ is $-\frac{3}{2}\ge n$

In interval form, solution set is $\left(-\infty ,-\frac{3}{2}\right]$

Solution set on number line is 

Step 1 - Write properties of inequality

Properties of inequalities are written in following table:

Step 2 - Simplify 12 1 4 − n 3 ≤ − 6 n

To simplify given inequality, first simplify parentheses then use properties of inequalities. Add same numbers from both sides does not change the sign of inequality. Finally use division property, divide both sides by same negative number reverse the sign of inequality as shown below

$\begin{array}{c}12\left(\frac{1}{4}-\frac{n}{3}\right)\le -6n\\ 3-4n\le -6n\\ 3-4n+4n\le -6n+4n\\ 3\le -2n\\ \frac{3}{-2}\ge \frac{-2n}{-2}\\ -\frac{3}{2}\ge n\end{array}$

Step 3 - Write solution set

The solution set of the inequality contains all real numbers less than or equal to $-\frac{3}{2}$

As, $-\frac{3}{2}$ is included in the set so bracket is used with it also $-\infty$ is not included so parentheses is used with it.

In interval form solution set is $\left(-\infty ,-\frac{3}{2}\right]$

Step 4 - Graph the solution set

As solution set of the inequality is all real numbers less than or equal to $-\frac{3}{2}$

So arrow is towards $-\infty$ and dot on $-\frac{3}{2}$ shows it is included in the solution set.