Math Snap

STEP 1

1. We are given the absolute value equation $|3x + 4| = 2x - 1$.
2. We need to solve for $x$.
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 $x$.
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 $|3x + 4| = 2x - 1$ implies two scenarios:
1. $3x + 4 = 2x - 1$
2. $3x + 4 = -(2x - 1)$

STEP 4

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

STEP 5

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

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