Question

1. |7x + 7| = 77
2. |6a + 9| + 1 = 4
3. -92 = -10|2 - 4n| + 8
4. 4v - 3 = |2v + 9|

Explanation:

Problem 13: $|7x + 7| = 77$

Step1: Split into two cases

Case 1: $7x + 7 = 77$; Case 2: $7x + 7 = -77$

Step2: Solve Case 1

Subtract 7 from both sides: $7x = 77 - 7 = 70$
Divide by 7: $x = \frac{70}{7} = 10$

Step3: Solve Case 2

Subtract 7 from both sides: $7x = -77 - 7 = -84$
Divide by 7: $x = \frac{-84}{7} = -12$

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Problem 15: $|6a + 9| + 1 = 4$

Step1: Isolate absolute value

Subtract 1 from both sides: $|6a + 9| = 4 - 1 = 3$

Step2: Split into two cases

Case 1: $6a + 9 = 3$; Case 2: $6a + 9 = -3$

Step3: Solve Case 1

Subtract 9: $6a = 3 - 9 = -6$
Divide by 6: $a = \frac{-6}{6} = -1$

Step4: Solve Case 2

Subtract 9: $6a = -3 - 9 = -12$
Divide by 6: $a = \frac{-12}{6} = -2$

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Problem 17: $-92 = -10|2 - 4n| + 8$

Step1: Isolate absolute value

Subtract 8: $-92 - 8 = -10|2 - 4n|$
Simplify: $-100 = -10|2 - 4n|$
Divide by -10: $|2 - 4n| = 10$

Step2: Split into two cases

Case 1: $2 - 4n = 10$; Case 2: $2 - 4n = -10$

Step3: Solve Case 1

Subtract 2: $-4n = 10 - 2 = 8$
Divide by -4: $n = \frac{8}{-4} = -2$

Step4: Solve Case 2

Subtract 2: $-4n = -10 - 2 = -12$
Divide by -4: $n = \frac{-12}{-4} = 3$

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Problem 19: $4v - 3 = |2v + 9|$

Step1: Split into two cases

Case 1: $4v - 3 = 2v + 9$; Case 2: $4v - 3 = -(2v + 9)$

Step2: Solve Case 1

Subtract $2v$, add 3: $4v - 2v = 9 + 3$
Simplify: $2v = 12$ → $v = \frac{12}{2} = 6$

Step3: Solve Case 2

Expand right side: $4v - 3 = -2v - 9$
Add $2v$, add 3: $4v + 2v = -9 + 3$
Simplify: $6v = -6$ → $v = \frac{-6}{6} = -1$

Step4: Verify solutions

For $v=6$: $4(6)-3=21$, $|2(6)+9|=21$ (valid)
For $v=-1$: $4(-1)-3=-7$, $|2(-1)+9|=7$ (valid, since $-7
eq 7$? Correction: Case 2 requires $4v-3 = -(2v+9)$, so $4v-3=-2v-9$, which gives $6v=-6$, $v=-1$. Check: $4(-1)-3=-7$, $|2(-1)+9|=7$. Since $-7
eq 7$, this solution is extraneous. Only $v=6$ is valid.

Answer:

1. $x = 10$ or $x = -12$
2. $a = -1$ or $a = -2$
3. $n = -2$ or $n = 3$
4. $v = 6$