\( \begin{array}{ll}\text { (a) } \frac{x}{3}-\frac{x}{2}>1 & \text { (b) } \frac{3 x}{4}-x \leq 1 \frac{1}{2} \\ \text { (c) } \frac{y+5}{3}+y \leq 1 & \text { (d) } \frac{3 y+2}{4}-\frac{y-6}{3}>0 \\ \text { (c) }(x+3)(x-4) \leq(x-3)(x+4) & \end{array} \)

Solve the inequality by following steps: - step0: Solve for \(x\): \(\frac{3x}{4}-x-\frac{3}{2}\leq 0\) - step1: Multiply both sides: \(\left(\frac{3x}{4}-x-\frac{3}{2}\right)\times 2\times 2\leq 0\times 2\times 2\) - step2: Multiply the terms: \(3x-4x-6\leq 0\) - step3: Simplify: \(-x-6\leq 0\) - step4: Move the constant to the right side: \(-x\leq 0+6\) - step5: Remove 0: \(-x\leq 6\) - step6: Change the signs: \(x\geq -6\) Solve the equation \( \frac{y+5}{3}+y-1<=0 \). Solve the inequality by following steps: - step0: Solve for \(y\): \(\frac{y+5}{3}+y-1\leq 0\) - step1: Multiply both sides: \(\left(\frac{y+5}{3}+y-1\right)\times 3\leq 0\times 3\) - step2: Multiply the terms: \(y+5+3y-3\leq 0\) - step3: Simplify: \(4y+2\leq 0\) - step4: Move the constant to the right side: \(4y\leq 0-2\) - step5: Remove 0: \(4y\leq -2\) - step6: Divide both sides: \(\frac{4y}{4}\leq \frac{-2}{4}\) - step7: Divide the numbers: \(y\leq -\frac{1}{2}\) Solve the equation \( (x+3)(x-4)-(x-3)(x+4)<=0 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(\left(x+3\right)\left(x-4\right)-\left(x-3\right)\left(x+4\right)\leq 0\) - step1: Simplify: \(-2x\leq 0\) - step2: Change the signs: \(2x\geq 0\) - step3: Rewrite the expression: \(x\geq 0\) Solve the equation \( \frac{x}{3}-\frac{x}{2}-1>0 \). Solve the inequality by following steps: - step0: Solve for \(x\): \(\frac{x}{3}-\frac{x}{2}-1>0\) - step1: Multiply both sides: \(\left(\frac{x}{3}-\frac{x}{2}-1\right)\times 3\times 2>0\times 3\times 2\) - step2: Multiply the terms: \(2x-3x-6>0\) - step3: Simplify: \(-x-6>0\) - step4: Move the constant to the right side: \(-x>0+6\) - step5: Remove 0: \(-x>6\) - step6: Change the signs: \(x<-6\) Solve the equation \( \frac{3 y+2}{4}-\frac{y-6}{3}>0 \). Solve the inequality by following steps: - step0: Solve for \(y\): \(\frac{3y+2}{4}-\frac{y-6}{3}>0\) - step1: Multiply both sides: \(\left(\frac{3y+2}{4}-\frac{y-6}{3}\right)\times 4\times 3>0\times 4\times 3\) - step2: Multiply the terms: \(9y+6-4y+24>0\) - step3: Simplify: \(5y+30>0\) - step4: Move the constant to the right side: \(5y>0-30\) - step5: Remove 0: \(5y>-30\) - step6: Divide both sides: \(\frac{5y}{5}>\frac{-30}{5}\) - step7: Divide the numbers: \(y>-6\) Let's solve each inequality step by step: ### (a) \( \frac{x}{3} - \frac{x}{2} > 1 \) 1. **Combine the fractions**: \[ \frac{x}{3} - \frac{x}{2} = \frac{2x - 3x}{6} = \frac{-x}{6} \] So, the inequality becomes: \[ \frac{-x}{6} > 1 \] 2. **Multiply both sides by -6** (remember to reverse the inequality): \[ x < -6 \] ### (b) \( \frac{3x}{4} - x \leq 1 \frac{1}{2} \) 1. **Convert \(1 \frac{1}{2}\) to an improper fraction**: \[ 1 \frac{1}{2} = \frac{3}{2} \] 2. **Rewrite the inequality**: \[ \frac{3x}{4} - x \leq \frac{3}{2} \] 3. **Combine like terms**: \[ \frac{3x}{4} - \frac{4x}{4} \leq \frac{3}{2} \implies \frac{-x}{4} \leq \frac{3}{2} \] 4. **Multiply both sides by -4** (reverse the inequality): \[ x \geq -6 \] ### (c) \( \frac{y+5}{3} + y \leq 1 \) 1. **Combine the terms**: \[ \frac{y+5}{3} + y = \frac{y + 5 + 3y}{3} = \frac{4y + 5}{3} \] 2. **Rewrite the inequality**: \[ \frac{4y + 5}{3} \leq 1 \] 3. **Multiply both sides by 3**: \[ 4y + 5 \leq 3 \] 4. **Isolate \(y\)**: \[ 4y \leq -2 \implies y \leq -\frac{1}{2} \] ### (d) \( \frac{3y+2}{4} - \frac{y-6}{3} > 0 \) 1. **Combine the fractions**: \[ \frac{3y+2}{4} - \frac{y-6}{3} = \frac{(3y+2) \cdot 3 - (y-6) \cdot 4}{12} \] This simplifies to: \[ \frac{9y + 6 - 4y + 24}{12} = \frac{5y + 30}{12} \] 2. **Rewrite the inequality**: \[ \frac{5y + 30}{12} > 0 \] 3. **Multiply both sides by 12**: \[ 5y + 30 > 0 \] 4. **Isolate \(y\)**: \[ 5y > -30 \implies y > -6 \] ### (e) \( (x+3)(x-4) \leq (x-3)(x+4) \) 1. **Expand both sides**: \[ x^2 - x - 12 \leq x^2 + x - 12 \] 2. **Subtract \(x^2 - 12\) from both sides**: \[ -x \leq x \implies -2x \leq 0 \] 3. **Divide by -2** (reverse the inequality): \[ x \geq 0 \] ### Summary of Solutions: - (a) \( x < -6 \) - (b) \( x \geq -6 \) - (c) \( y \leq -\frac{1}{2} \) - (d) \( y > -6 \) - (e) \( x \geq 0 \)