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Math  /  Algebra

Questionx=−5 and x=6x=-5 \text { and } x=6
5 Multiple Choice 1 point
Solve the following absolute value equation.
5. ∣3x+4∣=2x−1|3 x+4|=2 x-1 x=−3/5x=-3 / 5 x=−5x=-5 No solution x=−5x=-5 and x=−3/5x=-3 / 5 6 Multiple Choice 1 point

Studdy Solution

STEP 1

1. We are given the absolute value equation ∣3x+4∣=2x−1 |3x + 4| = 2x - 1 .
2. We need to solve for x x .
3. We will consider the definition of absolute value and solve the equation accordingly.
4. We will check the solutions to ensure they satisfy the original equation.

STEP 2

1. Understand the definition of absolute value and set up two equations.
2. Solve each equation for x x .
3. Verify the solutions in the context of the original equation.
4. Determine the correct solution(s) from the given multiple-choice options.

STEP 3

Understand the definition of absolute value and set up two equations.
The absolute value equation ∣3x+4∣=2x−1 |3x + 4| = 2x - 1 implies two scenarios:
1. 3x+4=2x−1 3x + 4 = 2x - 1
2. 3x+4=−(2x−1) 3x + 4 = -(2x - 1)

STEP 4

Solve each equation for x x .
For the first equation: 3x+4=2x−1 3x + 4 = 2x - 1
Subtract 2x 2x from both sides: x+4=−1 x + 4 = -1
Subtract 4 from both sides: x=−5 x = -5
For the second equation: 3x+4=−(2x−1) 3x + 4 = -(2x - 1)
Distribute the negative sign: 3x+4=−2x+1 3x + 4 = -2x + 1
Add 2x 2x to both sides: 5x+4=1 5x + 4 = 1
Subtract 4 from both sides: 5x=−3 5x = -3
Divide by 5: x=−35 x = -\frac{3}{5}

STEP 5

Verify the solutions in the context of the original equation.
Check x=−5 x = -5 : ∣3(−5)+4∣=2(−5)−1 |3(-5) + 4| = 2(-5) - 1 ∣−15+4∣=−10−1 |-15 + 4| = -10 - 1 ∣−11∣=−11 |-11| = -11 This is not true since absolute value cannot be negative.
Check x=−35 x = -\frac{3}{5} : ∣3(−35)+4∣=2(−35)−1 |3(-\frac{3}{5}) + 4| = 2(-\frac{3}{5}) - 1 ∣−1.8+4∣=−1.2−1 |-1.8 + 4| = -1.2 - 1 ∣2.2∣=−2.2 |2.2| = -2.2 This is not true since absolute value cannot be negative.

STEP 6

Determine the correct solution(s) from the given multiple-choice options.
Since neither solution satisfies the original equation, the correct answer is: "No solution"

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