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Solve the following absolute value inequality:

[tex]\[ \frac{2|x-5|}{7} \geq 10 \][/tex]

What is the solution for [tex]\( x \)[/tex]?

[tex]\[
\begin{array}{l}
x \geq \\
x \leq
\end{array}
\][/tex]


Sagot :

To solve the absolute value inequality [tex]\( \frac{2|x-5|}{7} \geq 10 \)[/tex]:

First, isolate the absolute value expression:

[tex]\[ \frac{2|x-5|}{7} \geq 10 \][/tex]

Multiply both sides of the inequality by 7 to clear the denominator:

[tex]\[ 2|x-5| \geq 70 \][/tex]

Next, divide both sides by 2 to further isolate the absolute value:

[tex]\[ |x-5| \geq 35 \][/tex]

The absolute value inequality [tex]\( |x-5| \geq 35 \)[/tex] can be interpreted as two separate inequalities:

[tex]\[ x - 5 \geq 35 \quad \text{or} \quad x - 5 \leq -35 \][/tex]

Solve each inequality separately:

1. [tex]\( x - 5 \geq 35 \)[/tex]:

Add 5 to both sides:

[tex]\[ x \geq 40 \][/tex]

2. [tex]\( x - 5 \leq -35 \)[/tex]:

Add 5 to both sides:

[tex]\[ x \leq -30 \][/tex]

Thus, the solution to the inequality [tex]\( \frac{2|x-5|}{7} \geq 10 \)[/tex] consists of two parts:

[tex]\[ x \geq 40 \quad \text{or} \quad x \leq -30 \][/tex]

To summarize, the positive absolute value for [tex]\( x \)[/tex] is:

[tex]\[ \begin{array}{l} x \geq 40 \\ x \leq -30 \end{array} \][/tex]