Question

Solve each inequality. Check your solution.

$\mathbf{-}\mathbf{2}\mathbf{(}\mathit{x}\mathbf{-}\mathbf{4}\mathbf{)}\mathbf{>}\mathbf{5}\mathit{x}\mathbf{-}\mathbf{13}$

Short answer

The solution of the given inequality$-2\left(x-4\right)>5x-13$ is $\mathit{x}\mathbf{<}\mathbf{3}$.

Step-by-step solution

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

The addition property of inequalities states that if the same number is added to each side of a true inequality, the resulting inequality is also true that is:

1. If $a>b$, then $a+c>b+c$.
2. 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:

1. If$a>b$ and$c$ is a positive number then $\frac{a}{c}>\frac{b}{c}$.
2. If$a<b$ and$c$ is a positive number then $\frac{a}{c}<\frac{b}{c}$.
3. If$a>b$ and$c$ is a negative number then $\frac{a}{c}<\frac{b}{c}$.
4. If$a<b$ and$c$ is a negative number then $\frac{a}{c}>\frac{b}{c}$.

Step 2. Solve the given inequality −2(x−4)>5x−13.

The solution of the given inequality$-2\left(x-4\right)>5x-13$ is:

$\begin{array}{c}-2\left(x-4\right)>5x-13\\ -2x+8>5x-13\left(\text{byusingdistributiveproperty}\right)\\ -2x+8+2x>5x-13+2x\left(\text{byusingadditionpropertyofinequality}\right)\\ -2x+2x+8>5x+2x-13\\ 8>7x-13\\ 8+13>7x-13+13\left(\text{byusingadditionpropertyofinequality}\right)\\ 21>7x\\ \frac{21}{7}\text{>}\frac{7x}{7}\left(\text{byusingdivisionpropertyofinequality}\right)\\ 3>x\\ x<3\end{array}$

Therefore, the solution of the given inequality$-2\left(x-4\right)>5x-13$ is $x<3$.

Step 3. Check the solution.

To perform the check of the solution, substitute a number less than 3 as $x<3$in the given inequality $-2\left(x-4\right)>5x-13$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let $x=2$, as $2<3$.

Now substitute 2 for $x$in the inequality $-2\left(x-4\right)>5x-13$.

$\begin{array}{c}-2\left(x-4\right)>5x-13\\ -2\left(2-4\right)>5\left(2\right)-13\\ -2\left(-2\right)>10-13\\ 4>-3\end{array}$

As, the condition obtained$4>-3$ is true, therefore the solution is correct.