Question

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

The sum of a number and$-2$ is no more than 6.

Short answer

The variable is x is and the inequality equation is $x-2\le 6$. The solution for the inequality is $x\le 8$.

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. Write the inequality for the given condition.

Let the number be x.

The sum of a number and$-2$ is no more than 6:

$x-2\le 6$

Step 3. Solve for x.

Solve for xin the equation $x-2\le 6$

$$\begin{gathered} x-2\le 6 \\ x-2+2\le 6+2 \\ x\le 8 \end{gathered}$$

Step 4. Check the solution when x=5.

Put the value of xas 5.

$$\begin{gathered} x-2\le 6 \\ 5-2\le 6 \\ 3\le 6\left[\text{TrueStatement}\right] \end{gathered}$$

Thus, the solution of the inequality equation $x-2\le 6$is $x\le 8$.