Q26. - Algebra 1

Question

Define a variable, write an inequality, and solve each problem. Check your solution. 

Two thirds of a number plus 5 is greater than 17.

Step-by-Step Solution

Answer

The variable is x is and the inequality equation is $\frac{2}{3}x+5>17$. The solution for the inequality is $x>18$.

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:

width="189" style="max-width: none; vertical-align: -38px;" $$\begin{gathered} 1.b\le c\Rightarrow b\pm a\le c\pm a \\ \begin{array}{l}2.b\le c\Rightarrow ab\le ac,\text{ if }a>0\\ 3.b\le c\Rightarrow ab\ge ac,\text{ if }a<0\end{array} \end{gathered}$$

Step 2. Write the inequality for the given condition.

Let the number be x.

Two thirds of a number plus 5 is greater than 17: 

$\frac{2}{3}x+5>17$ 

Step 3. Solve for x.

Solve for x  in the inequation $\frac{2}{3}x+5>17$

$$\begin{gathered} \frac{2}{3}x+5>17 \\ \frac{2}{3}x+5-5>17-5 \\ \frac{2}{3}x>12 \\ \frac{2}{3}x\times 3>12\times 3 \\ 2x>36 \\ \frac{2x}{2}>\frac{36}{2} \\ x>18 \end{gathered}$$

Step 4. Check the solution .

Put the value of x as 20

$$\begin{gathered} \frac{2}{3}x+5>17 \\ \frac{2}{3}\left(21\right)+5>17 \\ 14+5>17 \\ 19>17\left[\text{True Statement}\right] \end{gathered}$$

Thus, the solution of the inequality equation $\frac{2}{3}x+5>17$ is $x>18$.