Q5. - Algebra 2

Question

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

5. $5\ge 3x$

Step-by-Step Solution

Answer

Solution of inequality $5\ge 3x$ is $\frac{5}{3}\ge x$

In interval notation solution set is $\left(-\infty ,\frac{5}{3}\right]$

Solution set on line is 

Step 1 - Write properties of inequality

Properties of inequalities are written in following table:

Step 2 - Simplify 5 ≥ 3 x

To simplify given inequality, use division property of inequalities. Divide same positive number from both sides does not change inequality as shown below

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

Step 3 - Write solution set

The solution set of the inequality $\frac{5}{3}\ge x$ contains $\frac{5}{3}$ greater than or equal to all real number. 

In interval notation, solution set contains all real number between $-\infty$ to $\frac{5}{3}$ 

As $-\infty$ is not an end points so parentheses is used with it whereas $\frac{5}{3}$ is included in solution set so bracket is used with it.

Solution set is $\left(-\infty ,\frac{5}{3}\right]$ 

Step 4 - Graph the solution set

As solution set of the inequality $\frac{5}{3}\ge x$ contains  $\frac{5}{3}$ greater than or equal to all real number. 

So arrow is towards $-\infty$ and dot on $\frac{5}{3}$ denotes it is included in the solution set.