Question

which statements are true? check all that apply. the equation |-x - 4| = 8 will have two solutions. the equation 3.4|0.5x - 42.1| = -20.6 will have one solution. the equation |1/2 x - 3/4| = 0 will have no solutions. the equation |2x - 10| = -20 will have two solutions. the equation |0.5x - 0.75| + 4.6 = 0.25 will have no solutions. the equation |1/8 x - 1| = 5 will have infinitely many solutions.

Explanation:

Step1: Recall absolute - value property

The absolute - value of a number \(|a|\) is defined as \(|a|=

$$\begin{cases}a, & a\geq0\\-a, & a < 0\end{cases}$$

\), and \(|a|\geq0\) for all real numbers \(a\).

Step2: Analyze \(|-x - 4| = 8\)

We can rewrite it as \(-x - 4=8\) or \(-x - 4=-8\). Solving \(-x - 4 = 8\) gives \(-x=12\) or \(x=-12\), and solving \(-x - 4=-8\) gives \(-x=-4\) or \(x = 4\). So it has two solutions.

Step3: Analyze \(3.4|0.5x - 42.1|=-20.6\)

Since the left - hand side \(3.4|0.5x - 42.1|\geq0\) (because \(3.4>0\) and \(|0.5x - 42.1|\geq0\)) and the right - hand side \(-20.6<0\), there are no solutions.

Step4: Analyze \(|\frac{1}{2}x-\frac{3}{4}| = 0\)

We have \(\frac{1}{2}x-\frac{3}{4}=0\), which gives \(\frac{1}{2}x=\frac{3}{4}\) or \(x=\frac{3}{2}\). It has one solution.

Step5: Analyze \(|2x - 10|=-20\)

Since \(|2x - 10|\geq0\) and \(-20<0\), there are no solutions.

Step6: Analyze \(|0.5x - 0.75|+4.6 = 0.25\)

Rewrite it as \(|0.5x - 0.75|=0.25 - 4.6=-4.35\). Since \(|0.5x - 0.75|\geq0\) and \(-4.35<0\), there are no solutions.

Step7: Analyze \(|\frac{1}{8}x - 1| = 5\)

We can rewrite it as \(\frac{1}{8}x - 1 = 5\) or \(\frac{1}{8}x - 1=-5\). Solving \(\frac{1}{8}x - 1 = 5\) gives \(\frac{1}{8}x=6\) or \(x = 48\), and solving \(\frac{1}{8}x - 1=-5\) gives \(\frac{1}{8}x=-4\) or \(x=-32\). It has two solutions.

Answer:

The equation \(|-x - 4| = 8\) will have two solutions.
The equation \(|0.5x - 0.75|+4.6 = 0.25\) will have no solutions.