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Answer :
Let's solve the equation [tex]\(\left|\frac{2}{5} x + 1\right| - 7 = 0\)[/tex] step by step.
1. Isolate the Absolute Value:
Start by adding 7 to both sides of the equation to isolate the absolute value.
[tex]\[
\left|\frac{2}{5} x + 1\right| = 7
\][/tex]
2. Set Up Two Cases:
The absolute value equation [tex]\(\left|A\right| = B\)[/tex] can be split into two separate equations:
[tex]\[
\frac{2}{5} x + 1 = 7 \quad \text{and} \quad \frac{2}{5} x + 1 = -7
\][/tex]
3. Solve the First Equation:
[tex]\(\frac{2}{5} x + 1 = 7\)[/tex]
- Subtract 1 from both sides:
[tex]\[
\frac{2}{5} x = 6
\][/tex]
- Multiply both sides by [tex]\(\frac{5}{2}\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 6 \times \frac{5}{2} = 15
\][/tex]
4. Solve the Second Equation:
[tex]\(\frac{2}{5} x + 1 = -7\)[/tex]
- Subtract 1 from both sides:
[tex]\[
\frac{2}{5} x = -8
\][/tex]
- Multiply both sides by [tex]\(\frac{5}{2}\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -8 \times \frac{5}{2} = -20
\][/tex]
Therefore, the solutions to the equation [tex]\(\left|\frac{2}{5} x + 1\right| - 7 = 0\)[/tex] are [tex]\(x = -20\)[/tex] and [tex]\(x = 15\)[/tex]. The correct answer is: [tex]\(x = -20\)[/tex] and [tex]\(x = 15\)[/tex].
1. Isolate the Absolute Value:
Start by adding 7 to both sides of the equation to isolate the absolute value.
[tex]\[
\left|\frac{2}{5} x + 1\right| = 7
\][/tex]
2. Set Up Two Cases:
The absolute value equation [tex]\(\left|A\right| = B\)[/tex] can be split into two separate equations:
[tex]\[
\frac{2}{5} x + 1 = 7 \quad \text{and} \quad \frac{2}{5} x + 1 = -7
\][/tex]
3. Solve the First Equation:
[tex]\(\frac{2}{5} x + 1 = 7\)[/tex]
- Subtract 1 from both sides:
[tex]\[
\frac{2}{5} x = 6
\][/tex]
- Multiply both sides by [tex]\(\frac{5}{2}\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 6 \times \frac{5}{2} = 15
\][/tex]
4. Solve the Second Equation:
[tex]\(\frac{2}{5} x + 1 = -7\)[/tex]
- Subtract 1 from both sides:
[tex]\[
\frac{2}{5} x = -8
\][/tex]
- Multiply both sides by [tex]\(\frac{5}{2}\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -8 \times \frac{5}{2} = -20
\][/tex]
Therefore, the solutions to the equation [tex]\(\left|\frac{2}{5} x + 1\right| - 7 = 0\)[/tex] are [tex]\(x = -20\)[/tex] and [tex]\(x = 15\)[/tex]. The correct answer is: [tex]\(x = -20\)[/tex] and [tex]\(x = 15\)[/tex].
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