Question

1. -5|2x - 2| + 10 = -10
2. -9 - 4|v - 2| = -37
3. 5 - \frac{2}{3}|x + 4| = -11
4. \frac{|x|-5}{3}=2
5. -3|x - 5| = 18
6. -3|x - 5| = -18

Explanation:

Step1: Isolate the absolute - value term for equation 15

Starting with \(-5|2x - 2|+10=-10\), first subtract 10 from both sides: \(-5|2x - 2|=-10 - 10=-20\). Then divide both sides by - 5, we get \(|2x - 2| = 4\).

Step2: Solve for \(x\) in \(|2x - 2| = 4\)

Case 1: \(2x-2 = 4\), then \(2x=4 + 2=6\), and \(x = 3\).
Case 2: \(2x-2=-4\), then \(2x=-4 + 2=-2\), and \(x=-1\).

Step3: Isolate the absolute - value term for equation 16

Starting with \(-9-4|v - 2|=-37\), first add 9 to both sides: \(-4|v - 2|=-37 + 9=-28\). Then divide both sides by - 4, we get \(|v - 2| = 7\).

Step4: Solve for \(v\) in \(|v - 2| = 7\)

Case 1: \(v - 2=7\), then \(v=7 + 2=9\).
Case 2: \(v - 2=-7\), then \(v=-7 + 2=-5\).

Step5: Isolate the absolute - value term for equation 17

Starting with \(5-\frac{2}{3}|x + 4|=-11\), first subtract 5 from both sides: \(-\frac{2}{3}|x + 4|=-11 - 5=-16\). Then multiply both sides by \(-\frac{3}{2}\), we get \(|x + 4| = 24\).

Step6: Solve for \(x\) in \(|x + 4| = 24\)

Case 1: \(x + 4=24\), then \(x=24 - 4=20\).
Case 2: \(x + 4=-24\), then \(x=-24 - 4=-28\).

Step7: Isolate the absolute - value term for equation 18

Starting with \(\frac{|x|-5}{3}=2\), first multiply both sides by 3: \(|x|-5 = 6\). Then add 5 to both sides, we get \(|x|=11\).

Step8: Solve for \(x\) in \(|x| = 11\)

Case 1: \(x = 11\).
Case 2: \(x=-11\).

Step9: Analyze equation 19

Starting with \(-3|x - 5|=18\), divide both sides by - 3, we get \(|x - 5|=-6\). Since the absolute - value of a number is non - negative, there is no solution.

Step10: Solve for \(x\) in equation 20

Starting with \(-3|x - 5|=-18\), divide both sides by - 3, we get \(|x - 5| = 6\).
Case 1: \(x - 5=6\), then \(x=6 + 5=11\).
Case 2: \(x - 5=-6\), then \(x=-6 + 5=-1\).

Answer:

1. \(x = 3\) or \(x=-1\)
2. \(v = 9\) or \(v=-5\)
3. \(x = 20\) or \(x=-28\)
4. \(x = 11\) or \(x=-11\)
5. No solution
6. \(x = 11\) or \(x=-1\)