Let's solve the equation [tex]\( |5x - 3| = |x + 7| \)[/tex] step-by-step.
First, recall that for any two real numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex], [tex]\( |a| = |b| \)[/tex] implies that either [tex]\(a = b\)[/tex] or [tex]\(a = -b\)[/tex]. We will apply this property to solve the given equation.
### Case 1: [tex]\( 5x - 3 = x + 7 \)[/tex]
1. Start by setting the expressions inside the absolute values equal to each other:
[tex]\[
5x - 3 = x + 7
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[
5x - 3 = x + 7
\][/tex]
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
4x - 3 = 7
\][/tex]
Add 3 to both sides:
[tex]\[
4x = 10
\][/tex]
Divide both sides by 4:
[tex]\[
x = \frac{10}{4} = \frac{5}{2}
\][/tex]
### Case 2: [tex]\( 5x - 3 = -(x + 7) \)[/tex]
1. Set the expression inside the first absolute value equal to the negation of the expression inside the second absolute value:
[tex]\[
5x - 3 = -(x + 7)
\][/tex]
2. Simplify the right-hand side:
[tex]\[
5x - 3 = -x - 7
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
5x - 3 = -x - 7
\][/tex]
Add [tex]\(x\)[/tex] to both sides:
[tex]\[
6x - 3 = -7
\][/tex]
Add 3 to both sides:
[tex]\[
6x = -4
\][/tex]
Divide both sides by 6:
[tex]\[
x = \frac{-4}{6} = \frac{-2}{3}
\][/tex]
### Conclusion
The solutions to the equation [tex]\( |5x - 3| = |x + 7| \)[/tex] are:
[tex]\[
x = \frac{5}{2} \quad \text{and} \quad x = \frac{-2}{3}
\][/tex]
