Question

Solve the following inequation.

\[\frac{11 + 3x}{5} \geq 3 - x > -\frac{3}{2}, x \in \mathrm{R}.\] 

1. Write the solution set. 
2. Represent the solution set on the number line.

Answer 1

Step 1: Solve the given inequation

The given inequation is \[\frac{11 + 3x}{5} \geq 3 - x > -\frac{3}{2}, x \in \mathrm{R}\]

Splitting the inequation into two parts:

Part I:

\[\frac{11 + 3x}{5} \geq 3 - x\]

Multiplying both sides by 5:

\[11 + 3x \geq 5(3 - x)\] 

\[11 + 3x \geq 15 - 5x\] 

\[3x + 5x \geq 15 - 11\] 

\[8x \geq 4\] 

\[x \geq \frac{4}{8}\] 

\[x \geq \frac{1}{2} \quad \text{or} \quad x \geq 0.5\]

Part II:

\[3 - x > -\frac{3}{2}\] 

\[3 + \frac{3}{2} > x\] 

\[\frac{9}{2} > x\] 

\[x < \frac{9}{2} \quad \text{or} \quad x < 4.5\]

Combining Part I and Part II:

\[\frac{1}{2} \leq x < \frac{9}{2} \quad \text{or} \quad 0.5 \leq x < 4.5\]

a. Solution Set:

Solution Set = `x ∈ R : 1/2 x < 9/2`

\[\text{or} \quad {x \in \mathrm{R} : 0.5 \leq x < 4.5}\]

b. Representation on the Number Line:

Plot a darkened (closed) circle at \[x = \frac{1}{2}\] or (0.5) to indicate that it is included in the set.

Plot a hollow (open) circle at \[x = \frac{9}{2}\] or (4.5) to indicate that it is excluded from the set.

Draw a thick, darkened line segment connecting (0.5) and (4.5).