Question

which equations have no solution? choose three correct answers.
$-|-x| = -2$
$|x| = -15$
$-|x| = 12$
$-|x| = 0$
$-|-x| = 9$

Explanation:

Step1: Analyze \(|-x|=-2\)

The absolute value of a number is always non - negative, i.e., \(|a|\geq0\) for any real number \(a\). For the equation \(-|-x|=-2\), we can multiply both sides by \(- 1\) to get \(|-x| = 2\). Since \(|-x|=|x|\) and \(|x| = 2\) has solutions \(x = 2\) or \(x=-2\), this equation has solutions.

Step2: Analyze \(|x|=-15\)

By the definition of absolute value, \(|x|\) represents the distance of \(x\) from \(0\) on the number line. Distance is always a non - negative quantity, so \(|x|\geq0\) for all real numbers \(x\). Since \(- 15<0\), the equation \(|x|=-15\) has no solution.

Step3: Analyze \(-|x| = 12\)

We can rewrite this equation as \(|x|=-12\) (by multiplying both sides by \(-1\)). Since the absolute value of a number is non - negative (\(|x|\geq0\)) and \(-12 < 0\), the equation \(-|x|=12\) (or \(|x|=-12\)) has no solution.

Step4: Analyze \(-|x|=0\)

We can rewrite this equation as \(|x| = 0\) (by multiplying both sides by \(-1\)). The equation \(|x| = 0\) has a solution \(x = 0\).

Step5: Analyze \(-|-x|=9\)

First, note that \(|-x|=|x|\geq0\), so \(-|-x|\leq0\). But the right - hand side of the equation is \(9>0\). So the equation \(-|-x| = 9\) has no solution.

Answer:

The equations with no solution are \(|x|=-15\), \(-|x| = 12\), \(-|-x|=9\)