Question

1. |x - 3|+4x≤9

Explanation:

Step1: Consider two cases for the absolute - value

Case 1: When $x - 3\geq0$ (i.e., $x\geq3$), $|x - 3|=x - 3$.
The inequality becomes $(x - 3)+4x\leq9$.

Step2: Simplify the inequality in Case 1

Combine like - terms: $x-3 + 4x\leq9$ gives $5x-3\leq9$.
Add 3 to both sides: $5x\leq9 + 3$, so $5x\leq12$.
Divide both sides by 5: $x\leq\frac{12}{5}=2.4$. But this contradicts the condition $x\geq3$, so there is no solution in this case.

Step3: Consider the second case for the absolute - value

Case 2: When $x - 3\lt0$ (i.e., $x\lt3$), $|x - 3|=3 - x$.
The inequality becomes $(3 - x)+4x\leq9$.

Step4: Simplify the inequality in Case 2

Combine like - terms: $3 - x+4x\leq9$, which simplifies to $3 + 3x\leq9$.
Subtract 3 from both sides: $3x\leq9 - 3$, so $3x\leq6$.
Divide both sides by 3: $x\leq2$. This satisfies the condition $x\lt3$.

Answer:

$x\leq2$