Question

solve the inequality and express the solution set on the real number line.\\(\frac{x}{8} leq \frac{x}{2} - \frac{2x + 3}{4}\\)

Explanation:

Step1: Eliminate denominators

Multiply all terms by 8 (LCD of 8,2,4):
$$8\times\frac{x}{8} \leq 8\times\frac{x}{2} - 8\times\frac{2x+3}{4}$$
Simplify to get:
$$x \leq 4x - 2(2x+3)$$

Step2: Expand the right-hand side

Distribute the -2 to the terms in parentheses:
$$x \leq 4x - 4x - 6$$

Step3: Simplify like terms

Combine the x-terms on the right:
$$x \leq -6$$

Step4: Represent on number line

Draw a closed circle at -6 (since inequality is $\leq$) and shade all real numbers to the left of -6.

Answer:

The solution to the inequality is $\boldsymbol{x \leq -6}$.

To represent this on the real number line:

• Place a filled (closed) dot at $-6$.
• Draw a shaded line extending infinitely to the left from $-6$.