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Math  /  Algebra

QuestionSolve each equation. 26) ∣6x−4∣+3=19|6 x-4|+3=19 A) No solution. B) {−2,13}\left\{-2, \frac{1}{3}\right\} C) {103,−2}\left\{\frac{10}{3},-2\right\} D) {103}\left\{\frac{10}{3}\right\} 28) 5−2∣−6n+2∣=255-2|-6 n+2|=25 A) {−4}\{-4\} B) {−4,10}\{-4,10\} C) {195,−3}\left\{\frac{19}{5},-3\right\} D) No solution. 30) ∣b−4∣−4=2|b-4|-4=2 A) {10}\{10\} C) 4}\} B) {4,6}\{4,6\} D) {10,−2}\{10,-2\}
Solve each system.

Studdy Solution

STEP 1

1. For each equation involving absolute values, we will consider the definition of absolute value and split the equation into cases based on the argument inside the absolute value.
2. We will solve each case separately and check if the solutions satisfy the original equation.
3. The solutions will be collected and verified before selecting the correct option from the given choices.

STEP 2

1. Solve the equation ∣6x−4∣+3=19|6x - 4| + 3 = 19.
2. Solve the equation 5−2∣−6n+2∣=255 - 2|-6n + 2| = 25.
3. Solve the equation ∣b−4∣−4=2|b - 4| - 4 = 2.

STEP 3

First, isolate the absolute value in the equation ∣6x−4∣+3=19|6x - 4| + 3 = 19.
∣6x−4∣+3=19 |6x - 4| + 3 = 19 Subtract 3 from both sides:
∣6x−4∣=16 |6x - 4| = 16

STEP 4

Consider the two cases for the absolute value ∣A∣=B|A| = B:
1. 6x−4=166x - 4 = 16
2. 6x−4=−166x - 4 = -16
For the first case, solve 6x−4=166x - 4 = 16:
6x−4=16 6x - 4 = 16 Add 4 to both sides:
6x=20 6x = 20 Divide by 6:
x=206=103 x = \frac{20}{6} = \frac{10}{3}

STEP 5

For the second case, solve 6x−4=−166x - 4 = -16:
6x−4=−16 6x - 4 = -16 Add 4 to both sides:
6x=−12 6x = -12 Divide by 6:
x=−2 x = -2

STEP 6

The solutions for ∣6x−4∣+3=19|6x - 4| + 3 = 19 are x=103x = \frac{10}{3} and x=−2x = -2.
Thus, the solution set is: {103,−2} \left\{ \frac{10}{3}, -2 \right\}

STEP 7

First, isolate the absolute value in the equation 5−2∣−6n+2∣=255 - 2|-6n + 2| = 25.
5−2∣−6n+2∣=25 5 - 2|-6n + 2| = 25 Subtract 5 from both sides:
−2∣−6n+2∣=20 -2|-6n + 2| = 20 Divide by -2:
∣−6n+2∣=−10 |-6n + 2| = -10
Since the absolute value cannot be negative, this equation has no solution. Therefore, the solution set is:
No solution \text{No solution}

STEP 8

First, isolate the absolute value in the equation ∣b−4∣−4=2|b - 4| - 4 = 2.
∣b−4∣−4=2 |b - 4| - 4 = 2 Add 4 to both sides:
∣b−4∣=6 |b - 4| = 6

STEP 9

Consider the two cases for the absolute value ∣A∣=B|A| = B:
1. b−4=6b - 4 = 6
2. b−4=−6b - 4 = -6
For the first case, solve b−4=6b - 4 = 6:
b−4=6 b - 4 = 6 Add 4 to both sides:
b=10 b = 10

STEP 10

For the second case, solve b−4=−6b - 4 = -6:
b−4=−6 b - 4 = -6 Add 4 to both sides:
b=−2 b = -2

STEP 11

The solutions for ∣b−4∣−4=2|b - 4| - 4 = 2 are b=10b = 10 and b=−2b = -2.
Thus, the solution set is: {10,−2} \left\{ 10, -2 \right\}
The solutions to the equations are:
1. ∣6x−4∣+3=19|6x - 4| + 3 = 19: {103,−2}\left\{\frac{10}{3}, -2\right\} (Option C)
2. 5−2∣−6n+2∣=255 - 2|-6n + 2| = 25: No solution (Option D)
3. ∣b−4∣−4=2|b - 4| - 4 = 2: {10,−2}\left\{ 10, -2 \right\} (Option D)

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