Question

solve the following absolute value inequality.
\\(\frac{2|x - 5|}{7} \geq 10\\)
\\(x \geq ?\\) or \\(x \leq\\)

Explanation:

Step1: Eliminate the denominator

Multiply both sides of the inequality \(\frac{2|x - 5|}{7} \geq 10\) by \(7\) to get rid of the denominator. Since \(7\) is positive, the direction of the inequality sign remains the same.

$$ 2|x - 5| \geq 7\times10 $$

$$ 2|x - 5| \geq 70 $$

Step2: Isolate the absolute value

Divide both sides of the inequality \(2|x - 5| \geq 70\) by \(2\) to isolate the absolute value expression.

$$ |x - 5| \geq \frac{70}{2} $$

$$ |x - 5| \geq 35 $$

Step3: Solve the absolute value inequality

The absolute value inequality \(|x - 5| \geq 35\) means that \(x - 5 \geq 35\) or \(x - 5 \leq - 35\) (because if \(|a| \geq b\) (\(b>0\)), then \(a \geq b\) or \(a \leq -b\)).

For the first case \(x - 5 \geq 35\):
Add \(5\) to both sides of the inequality.

$$ x \geq 35 + 5 $$

$$ x \geq 40 $$

For the second case \(x - 5 \leq - 35\):
Add \(5\) to both sides of the inequality.

$$ x \leq - 35 + 5 $$

$$ x \leq - 30 $$

Answer:

\(x \geq 40\) or \(x \leq - 30\) (For the blank in \(x \geq [?]\), the answer is \(40\))