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Determine the solutions of the equation:

[tex]\left|\frac{2}{5} x+1\right|-7=0[/tex]

A. [tex]x = -30[/tex] and [tex]x = 15[/tex]
B. [tex]x = -20[/tex] and [tex]x = 15[/tex]
C. [tex]x = -20[/tex] and [tex]x = 20[/tex]
D. [tex]x = -15[/tex] and [tex]x = 15[/tex]

Answer :

Sure! Let's solve the equation step-by-step:

The given equation is:

[tex]\[
\left|\frac{2}{5} x + 1\right| - 7 = 0
\][/tex]

1. Isolate the absolute value expression:

Add 7 to both sides:

[tex]\[
\left|\frac{2}{5} x + 1\right| = 7
\][/tex]

2. Remove the absolute value by considering two cases:

The expression inside the absolute value can be equal to 7 or -7.

Case 1: [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]\(x = 6 \times \frac{5}{2}\)[/tex]

[tex]\(x = 15\)[/tex]

Case 2: [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]\(x = -8 \times \frac{5}{2}\)[/tex]

[tex]\(x = -20\)[/tex]

3. Write the solutions:

The solutions to the equation [tex]\(\left|\frac{2}{5} x + 1\right| - 7 = 0\)[/tex] are [tex]\(x = 15\)[/tex] and [tex]\(x = -20\)[/tex].

Thus, the correct answer is [tex]\(x = -20\)[/tex] and [tex]\(x = 20\)[/tex].

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