Example Question - solve inequality

Solving a Compound Inequality

<p>Given the compound inequality \( \frac{3x+1}{2} \leq 5 + 2x \) and \( 1 + 5x \geq 8x \).</p> <p>First, solve \( \frac{3x+1}{2} \leq 5 + 2x \):</p> <p>Multiply both sides by 2 to eliminate the fraction: \( 3x+1 \leq 10 + 4x \).</p> <p>Subtract \( 3x \) from both sides: \( 1 \leq 10 + x \).</p> <p>Subtract 10 from both sides: \( -9 \leq x \) or \( x \geq -9 \).</p> <p>Now, solve \( 1 + 5x \geq 8x \):</p> <p>Subtract \( 5x \) from both sides: \( 1 \geq 3x \).</p> <p>Divide both sides by 3: \( \frac{1}{3} \geq x \) or \( x \leq \frac{1}{3} \).</p> <p>Combine the solutions: \( x \geq -9 \) and \( x \leq \frac{1}{3} \).</p> <p>So the solution to the compound inequality is \( -9 \leq x \leq \frac{1}{3} \).</p> <p>Thus, the correct answer is D. \( -9 \leq x \leq \frac{1}{3} \).</p>