Question

absolute value equations worksheet (set 3)
name:
date:
directions:
solve each absolute value equation algebraically. then graph the solution(s) on a number line. identify whether the equation has one solution, two solutions, or no solution.

part a: solve algebraically & graph (1--12)

1. |x - 8| = 3
2. |2n + 4| = 10
3. |n + 7| = 5
4. |2n + 2| = 10
5. |n| = 14
6. |8 - n| = 4
7. |5x - 2| = 18
8. |x - 22| = 0
9. |4x + 3| = 9
10. |x + 1| = 6
11. |7x - 5| = 16
12. |x - 11| = 12

part b: special solutions (13--20)
determine whether the equation has no solution, one solution, or two solutions. solve when possible.

1. |x + 3| = -4
2. |5x - 8| = -12
3. |x - 2| = 0

Explanation:

Step1: Solve \(|x - 8| = 3\)

By the definition of absolute value, if \(|A| = B\) (\(B\geq0\)), then \(A = B\) or \(A = -B\). So for \(|x - 8| = 3\), we have two cases:
Case 1: \(x - 8 = 3\)
Adding 8 to both sides: \(x = 3 + 8 = 11\)
Case 2: \(x - 8 = -3\)
Adding 8 to both sides: \(x = -3 + 8 = 5\)
So the solutions are \(x = 5\) and \(x = 11\).

Step2: Solve \(|2m + 4| = 10\)

Using the absolute value definition:
Case 1: \(2m + 4 = 10\)
Subtract 4: \(2m = 10 - 4 = 6\)
Divide by 2: \(m = 3\)
Case 2: \(2m + 4 = -10\)
Subtract 4: \(2m = -10 - 4 = -14\)
Divide by 2: \(m = -7\)
Solutions: \(m = 3\) and \(m = -7\).

Step3: Solve \(|n + 7| = 5\)

Case 1: \(n + 7 = 5\)
Subtract 7: \(n = 5 - 7 = -2\)
Case 2: \(n + 7 = -5\)
Subtract 7: \(n = -5 - 7 = -12\)
Solutions: \(n = -2\) and \(n = -12\).

Step4: Solve \(|3n + 2| = 13\)

Case 1: \(3n + 2 = 13\)
Subtract 2: \(3n = 13 - 2 = 11\)
Divide by 3: \(n = \frac{11}{3}\)
Case 2: \(3n + 2 = -13\)
Subtract 2: \(3n = -13 - 2 = -15\)
Divide by 3: \(n = -5\)
Solutions: \(n = \frac{11}{3}\) and \(n = -5\).

Step5: Solve \(|6k| = 14\)

Case 1: \(6k = 14\)
Divide by 6: \(k = \frac{14}{6} = \frac{7}{3}\)
Case 2: \(6k = -14\)
Divide by 6: \(k = -\frac{14}{6} = -\frac{7}{3}\)
Solutions: \(k = \frac{7}{3}\) and \(k = -\frac{7}{3}\).

Step6: Solve \(|8 - x| = 4\)

Rewrite as \(|x - 8| = 4\) (since \(|a - b| = |b - a|\))
Case 1: \(x - 8 = 4\)
Add 8: \(x = 12\)
Case 2: \(x - 8 = -4\)
Add 8: \(x = 4\)
Solutions: \(x = 12\) and \(x = 4\).

Step7: Solve \(|5x - 2| = 18\)

Case 1: \(5x - 2 = 18\)
Add 2: \(5x = 20\)
Divide by 5: \(x = 4\)
Case 2: \(5x - 2 = -18\)
Add 2: \(5x = -16\)
Divide by 5: \(x = -\frac{16}{5}\)
Solutions: \(x = 4\) and \(x = -\frac{16}{5}\).

Step8: Solve \(|x - 220| = 0\)

By absolute value definition, \(|A| = 0\) implies \(A = 0\). So \(x - 220 = 0\)
Add 220: \(x = 220\)
One solution: \(x = 220\).

Step9: Solve \(|4x + 3| = 9\)

Case 1: \(4x + 3 = 9\)
Subtract 3: \(4x = 6\)
Divide by 4: \(x = \frac{6}{4} = \frac{3}{2}\)
Case 2: \(4x + 3 = -9\)
Subtract 3: \(4x = -12\)
Divide by 4: \(x = -3\)
Solutions: \(x = \frac{3}{2}\) and \(x = -3\).

Step10: Solve \(|x + 7| = 6\)

Case 1: \(x + 7 = 6\)
Subtract 7: \(x = -1\)
Case 2: \(x + 7 = -6\)
Subtract 7: \(x = -13\)
Solutions: \(x = -1\) and \(x = -13\).

Step11: Solve \(|7k - 5| = 16\)

Case 1: \(7k - 5 = 16\)
Add 5: \(7k = 21\)
Divide by 7: \(k = 3\)
Case 2: \(7k - 5 = -16\)
Add 5: \(7k = -11\)
Divide by 7: \(k = -\frac{11}{7}\)
Solutions: \(k = 3\) and \(k = -\frac{11}{7}\).

Step12: Solve \(|n - 11| = 12\)

Case 1: \(n - 11 = 12\)
Add 11: \(n = 23\)
Case 2: \(n - 11 = -12\)
Add 11: \(n = -1\)
Solutions: \(n = 23\) and \(n = -1\).

Step13: Analyze \(|x + 3| = -4\)

The absolute value of a number is always non - negative (i.e., \(|A|\geq0\) for any real number \(A\)). Since the right - hand side is \(-4\) (negative), there is no solution.

Step14: Analyze \(|5x - 8| = -12\)

Again, the absolute value \(|5x - 8|\) is non - negative, and the right - hand side is \(-12\) (negative). So there is no solution.

Step15: Solve \(|x - 2| = 0\)

By the definition of absolute value, if \(|A| = 0\), then \(A = 0\). So \(x - 2 = 0\)
Add 2 to both sides: \(x = 2\)
One solution: \(x = 2\).

Answer:

1. \(x = 5\) or \(x = 11\)
2. \(m = 3\) or \(m = -7\)
3. \(n = -2\) or \(n = -12\)
4. \(n=\frac{11}{3}\) or \(n = - 5\)
5. \(k=\frac{7}{3}\) or \(k = -\frac{7}{3}\)
6. \(x = 4\) or \(x = 12\)
7. \(x = 4\) or \(x=-\frac{16}{5}\)
8. \(x = 220\)
9. \(x=\frac{3}{2}\) or \(x=-3\)
10. \(x=-1\) or \(x = - 13\)
11. \(k = 3\) or \(k=-\frac{11}{7}\)
12. \(n = 23\) or \(n=-1\)
13. No solution
14. No solution
15. \(x = 2\)