Solve the inequality $$ \frac{5(x+1)}{3}

Question

UpStudy AI · Accepted Answer

We start with the inequality:
$$
\frac{5(x+1)}{3} < x - \frac{3(1-x)}{5}
$$

**Step 1.** Multiply both sides by the common denominator $$15$$ (since $$15 > 0$$, the inequality direction is preserved):
$$
15 \cdot \frac{5(x+1)}{3} < 15 \cdot \left( x - \frac{3(1-x)}{5} \right)
$$
Simplify each term:
$$
15 \cdot \frac{5(x+1)}{3} = 25(x+1)
$$
$$
15 \cdot \left( x - \frac{3(1-x)}{5} \right) = 15x - 9(1-x)
$$
So the inequality becomes:
$$
25(x+1) < 15x - 9(1-x)
$$

**Step 2.** Expand both sides:
$$
25x + 25 < 15x - 9 + 9x
$$
Combine like terms on the right:
$$
25x + 25 < 24x - 9
$$

**Step 3.** Isolate $$x$$ by subtracting $$24x$$ from both sides:
$$
25x - 24x + 25 < -9 \quad \Longrightarrow \quad x + 25 < -9
$$
Subtract $$25$$ from both sides:
$$
x < -9 - 25 \quad \Longrightarrow \quad x < -34
$$

**Step 4.** Express the solution in interval notation:
$$
(-\infty, -34)
$$
The solution to the inequality is $$ x < -34 $$, which can be written in interval form as $$ (-\infty, -34) $$.

Solve the inequality and express the answer in interval form.

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