(2.2) Solve the following inequal and show number line solution. (2.2) \( \frac{3 x}{4}-x \leq 1 \frac{1}{2} \quad \) (3) \( \begin{array}{l}\text { (2.22) }-5 \leq 2 x+1 \leq 5 \\ \text { (2.3) Use Subs Methnd }\end{array} \)

To solve the first inequality \( \frac{3x}{4} - x \leq 1.5 \), first simplify the left side. Rewrite \(-x\) as \(-\frac{4x}{4}\), giving us: \[\frac{3x}{4} - \frac{4x}{4} \leq 1.5\] which simplifies to \(-\frac{x}{4} \leq 1.5\). Multiply both sides by -4 (remember to flip the inequality): \[x \geq -6.\] On the number line, this means shading everything to the right of -6, including -6 itself. For the compound inequality \(-5 \leq 2x + 1 \leq 5\), break it into two parts. Start with \(-5 \leq 2x + 1\): Subtract 1 to get \(-6 \leq 2x\), then divide by 2: \[-3 \leq x.\] Now for \(2x + 1 \leq 5\): Subtracting 1 gives us \(2x \leq 4\), and dividing by 2 results in: \[x \leq 2.\] So, combining these, we have \(-3 \leq x \leq 2\). On the number line, shade between -3 and 2, including both endpoints. Now, to use the substitution method for a more complex inequality scenario, consider defining a new variable, say \( y = 2x + 1 \). This transforms our inequality directly into a manageable form centered around a single variable, allowing you to address the two bounds independently and visualize their relation easily. Just remember, substituting can make solving more straightforward and less confusing!