Question

Solve each inequality. Check your solution.

$\frac{1}{3}y\ge 5$

Short answer

The required value of y is $\left(-\infty ,15\right]$.

Step-by-step solution

Step 1. State the concept for inequality.

To graph the endpoint with strict inequality < 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:

$$\begin{gathered} 1.b\le c\Rightarrow b\pm a\le c\pm a \\ 2.b\le c\Rightarrow ab\le ac,\text{if}a>0 \\ 3.b\le c\Rightarrow ab\ge ac,\text{if}a<0 \end{gathered}$$

Step 2. Solve the inequality for x.

In order to solve the inequality:

$\frac{1}{3}y\le 5$

Multiply both sides by 3, as

$$\begin{gathered} \frac{1}{3}y\le 5 \\ \frac{1}{3}y\times 3\le 5\times 3 \\ y\le 15 \end{gathered}$$

Thus, the inequality is true for y less than equal to 15, thus the solution of given inequality in interval form is, as $\left(-\infty ,15\right]$.

Step 3. Check the solution.

Now, to check the answer solution for the given inequality first check the solution at end point by converting it into an equation. Putting y equals to 15 in left hand side

$$\begin{gathered} \frac{15}{3}=5 \\ =\text{RightHandside} \end{gathered}$$

Now put any value any value of y less than 15, say 12 and check whether it satisfies the inequality, as

$\frac{12}{3}=4<5$

Since, both conditions are satisfied thus solution of inequality is correct.