2.2) Solve the following inequan and show number ine solution. (2.2) \( \frac{3 x}{4}-x \leq 1 \frac{1}{2} \) \( \begin{array}{l}\text { (222) }-5 \leq 2 x+1 \leq 5 \\ \text { (2.3) Use Subs Methnd }\end{array} \)

To solve the inequality \( \frac{3x}{4} - x \leq 1 \frac{1}{2} \), first convert \(1 \frac{1}{2}\) to an improper fraction: \(1 \frac{1}{2} = \frac{3}{2}\). Rewriting the left side gives \( \frac{3x}{4} - \frac{4x}{4} \leq \frac{3}{2} \), simplifying to \( -\frac{x}{4} \leq \frac{3}{2} \). Multiply both sides by -4 (remembering to flip the inequality): \( x \geq -6\). Now for the double inequality \(-5 \leq 2x + 1 \leq 5\), we can break it into two parts. First, solve \(-5 \leq 2x + 1\): subtract 1 from both sides to get \(-6 \leq 2x\), leading to \(x \geq -3\). Next, solve \(2x + 1 \leq 5\): subtract 1, yielding \(2x \leq 4\), which simplifies to \(x \leq 2\). Thus, combining both results gives \(-3 \leq x \leq 2\). Now shiny number line shows the solution as a closed interval from -3 to 2! 🌟 For a fun trivia twist, did you know inequalities can help in real life, like when budgeting or planning? They indicate what range of values meet your needs. So next time you read a recipe or manage a project, think of those delightful inequalities guiding your choices! Also, if you want to dive deeper, there are fantastic resources out there like Khan Academy or math-focused YouTube channels that explain inequalities with engaging visuals and examples. You might even find some quirky tutorials that make learning these concepts a blast!