Q27. - Algebra 1

Question

Solve each inequality. Check your solution.

$18\le -2x+8$

Step-by-Step Solution

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 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 &#8804; &#8722; 2 x + 8 .

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

$$\begin{gathered} 18\le -2x+8 \\ 18-8\le -2x+8-8\left(\text{by using subtraction property of inequality}\right) \\ 10\le -2x \\ \frac{10}{-2}\ge \frac{-2x}{-2}\left(\text{by using division property of inequality}\right) \\ -5\ge x \\ x\le -5 \end{gathered}$$

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.