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[tex]\[ \left|\frac{x-3}{5}\right| \ \textless \ 2 \][/tex]


Sagot :

Sure! Let's solve the inequality step-by-step.

Given inequality:
[tex]\[ \left|\frac{x-3}{5}\right| < 2 \][/tex]

The absolute value inequality [tex]\(\left| A \right| < B\)[/tex] can be split into two inequalities:
[tex]\[ - B < A < B \][/tex]

So, we can rewrite the given inequality as:
[tex]\[ -2 < \frac{x-3}{5} < 2 \][/tex]

This compound inequality can be split into two individual inequalities:
1. [tex]\(-2 < \frac{x-3}{5}\)[/tex]
2. [tex]\(\frac{x-3}{5} < 2\)[/tex]

Now, let's solve each part individually.

### Solving [tex]\(-2 < \frac{x-3}{5}\)[/tex]:

1. Multiply all parts of the inequality by 5 to get rid of the fraction:
[tex]\[ -2 \times 5 < \frac{x-3}{5} \times 5 \implies -10 < x - 3 \][/tex]

2. Add 3 to all parts of the inequality:
[tex]\[ -10 + 3 < x - 3 + 3 \implies -7 < x \][/tex]

Thus, one part of the inequality is:
[tex]\[ -7 < x \][/tex]

### Solving [tex]\(\frac{x-3}{5} < 2\)[/tex]:

1. Multiply all parts of the inequality by 5 to get rid of the fraction:
[tex]\[ \frac{x-3}{5} \times 5 < 2 \times 5 \implies x - 3 < 10 \][/tex]

2. Add 3 to all parts of the inequality:
[tex]\[ x - 3 + 3 < 10 + 3 \implies x < 13 \][/tex]

Thus, the other part of the inequality is:
[tex]\[ x < 13 \][/tex]

### Combining the results:
We combine the two results to form the compound inequality:
[tex]\[ -7 < x < 13 \][/tex]

### Interval Notation:
The solution set in interval notation is:
[tex]\[ (-7, 13) \][/tex]

So, the final solution to the given inequality [tex]\(\left|\frac{x-3}{5}\right| < 2\)[/tex] in interval notation is [tex]\((-7, 13)\)[/tex].