2. Знайдіть цілі розв'язки нерівності
[tex]0 < 1 + \frac{2 + 3x}{2} < 3[/tex]
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To solve the inequality tex0 < 1 + \frac{2 + 3x}{2} < 3/tex, we can break it down into two separate inequalities:

1) tex0 < 1 + \frac{2 + 3x}{2}/tex

2) tex1 + \frac{2 + 3x}{2} < 3/tex

Let's solve each inequality step by step:

1) tex0 < 1 + \frac{2 + 3x}{2}/tex

First, let's simplify the expression inside the fraction:

tex\frac{2 + 3x}{2}/tex

Now, let's multiply both sides of the inequality by 2 to eliminate the fraction:

tex0 \cdot 2 < 1 \cdot 2 + (2 + 3x)/tex

Simplifying further gives:

tex0 < 2 + 2 + 3x/tex

tex0 < 4 + 3x/tex

Next, let's isolate the variable by subtracting 4 from both sides:

tex0 - 4 < 4 + 3x - 4/tex

tex-4 < 3x/tex

Finally, divide both sides by 3 to solve for x:

tex\frac{-4}{3} < x/tex

2) tex1 + \frac{2 + 3x}{2} < 3/tex

Let's simplify the expression inside the fraction:

tex\frac{2 + 3x}{2}/tex

Now, multiply both sides of the inequality by 2 to eliminate the fraction:

tex1 \cdot 2 + \frac{2 + 3x}{2} \cdot 2 < 3 \cdot 2/tex

Simplifying further gives:

tex2 + 2 + 3x < 6/tex

tex4 + 3x < 6/tex

Next, isolate the variable by subtracting 4 from both sides:

tex4 + 3x - 4 < 6 - 4/tex

tex3x < 2/tex

Finally, divide both sides by 3 to solve for x:

texx < \frac{2}{3}/tex

Therefore, the solution to the given inequality is tex\frac{-4}{3} < x < \frac{2}{3}/tex.