Question

Solve each inequality. Check your solution.

$18\le -2x+8$

Short answer

The solution of the given inequality$18\le -2x+8$ is $x\le -5$.

Check:

To perform the check of the solution, substitute a number less than or equal to$-5$ as$x\le -5$in the given inequality $18\le -2x+8$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let $x=-6$, as $-6\le -5$.

Now substitute$-6$ for x in the inequality $18\le -2x+8$.

$$\begin{gathered} 18\le -2x+8 \\ 18\le -2\left(-6\right)+8 \\ 18\le 12+8 \\ 18\le 20 \end{gathered}$$

As, the condition obtained$18\le 20$ is true, therefore the solution is correct.

Step-by-step solution

Step 1. Write the subtraction and division property of inequalities. 

The subtraction property of inequalities states that if the same number is subtracted from each side of a true inequality, the resulting inequality is also true that is:

(i) If $a>b$, then $a-c>b-c$.

(ii) If $a<b$, then $a-c<b-c$.

The division property of inequalities states that if both sides of the inequality are divided by a positive number the sign of the inequality remains the same and if both sides of the inequality are divided by a negative number then the sign of the inequality changes that is:

(i) If$a>b$ and c is a positive number then $\frac{a}{c}>\frac{b}{c}$.

(ii) If$a<b$ and c is a positive number then $\frac{a}{c}<\frac{b}{c}$.

(iii) If$a>b$ and c is a negative number then $\frac{a}{c}<\frac{b}{c}$.

(iv) If$a<b$ and c is a negative number then $\frac{a}{c}>\frac{b}{c}$.

Step 2. Solve the given inequality 18≤−2x+8.

The solution of the given inequality$18\le -2x+8$ is:

$$\begin{gathered} 18\le -2x+8 \\ 18-8\le -2x+8-8\left(\text{byusingsubtractionpropertyofinequality}\right) \\ 10\le -2x \\ \frac{10}{-2}\ge \frac{-2x}{-2}\left(\text{byusingdivisionpropertyofinequality}\right) \\ -5\ge x \\ x\le -5 \end{gathered}$$

Therefore, the solution of the given inequality$18\le -2x+8$ is $x\le -5$.

Step 3. Check the solution.

To perform the check of the solution, substitute a number less than or equal to$-5$ as$x\le -5$ in the given inequality $18\le -2x+8$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let $x=-6$, as $-6\le -5$.

Now substitute$-6$ for x in the inequality $18\le -2x+8$.

$$\begin{gathered} 18\le -2x+8 \\ 18\le -2\left(-6\right)+8 \\ 18\le 12+8 \\ 18\le 20 \end{gathered}$$

As, the condition obtained$18\le 20$ is true, therefore the solution is correct.