Question

Solve the inequation given below. Write the solution set and represent it on the number line:

$$11x - 4 < 15x + 4 \leq 13x + 14, x \in \mathrm{W}$$

Answer 1

Given inequation: $$11x - 4 < 15x + 4 \leq 13x + 14, x \in \mathrm{W}$$

Step 1: Split into two separate inequations: $$11x - 4 < 15x + 4 \quad \text{and} \quad 15x + 4 \leq 13x + 14$$

Step 2: Solve the first inequation: $$11x - 4 < 15x + 4$$

$$\Rightarrow -4 - 4 < 15x - 11x$$

$$\Rightarrow -8 < 4x$$

$$\Rightarrow -2 < x \quad \text{or} \quad x > -2$$

Step 3: Solve the second inequation: $$15x + 4 \leq 13x + 14$$

$$\Rightarrow 15x - 13x \leq 14 - 4$$

$$\Rightarrow 2x \leq 10$$

$$\Rightarrow x \leq 5$$

Step 4: Combine the inequalities: $$-2 < x \leq 5$$

Step 5: Determine the solution set: Since $$x \in \mathrm{W}$$ (whole numbers: $${0, 1, 2, 3, \dots}$$), $$\text{Solution set} = {0, 1, 2, 3, 4, 5}$$

Representation on the number line: Mark bold darkened dots (solid circles) at the points $$0, 1, 2, 3, 4,\text{ and } 5$$ on the number line to represent each whole number in the solution set.