For what values of x does the absolute value equation |2x - 1| = 7 hold true?
Understand the Problem
The question asks us to find the values of x that satisfy the absolute value equation |2x - 1| = 7. This involves solving for x in two separate cases: when 2x - 1 is positive and when it is negative.
Answer
$x = 4, -3$
Answer for screen readers
$x = 4, -3$
Steps to Solve
- Set up two cases
Because we have an absolute value equation, we need to consider two cases: one where the expression inside the absolute value is positive or zero, and one where it is negative. This gives us two equations to solve.
- Solve the first case: $2x - 1 = 7$
If $2x - 1$ is positive or zero, then $|2x - 1| = 2x - 1$. So, we solve the equation $2x - 1 = 7$.
Add 1 to both sides: $2x - 1 + 1 = 7 + 1$ $2x = 8$
Divide both sides by 2: $\frac{2x}{2} = \frac{8}{2}$ $x = 4$
- Solve the second case: $2x - 1 = -7$
If $2x - 1$ is negative, then $|2x - 1| = -(2x - 1)$. So, we solve the equation $-(2x - 1) = 7$, which is equivalent to $2x - 1 = -7$.
Add 1 to both sides: $2x - 1 + 1 = -7 + 1$ $2x = -6$
Divide both sides by 2: $\frac{2x}{2} = \frac{-6}{2}$ $x = -3$
- Check the solutions
We need to check if both solutions, $x = 4$ and $x = -3$, satisfy the original equation.
For $x = 4$: $|2(4) - 1| = |8 - 1| = |7| = 7$. This solution is valid.
For $x = -3$: $|2(-3) - 1| = |-6 - 1| = |-7| = 7$. This solution is also valid.
$x = 4, -3$
More Information
Absolute value equations often have two solutions because the absolute value of a number is its distance from zero, which can be in either direction (positive or negative).
Tips
A common mistake is to only consider the positive case and forget to solve for the negative case of the absolute value expression. Always remember to set up and solve both possibilities to find all solutions.
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