Question

solving math inequality: remember to reverse the inequality sign whenever you multiply or divide both sides by a negative number. -3x + 5 > 22 - 5 -3x > 17 -3x / -3 < 17 / -3 (dividing by -3 means i have to reverse >) x < -17 / 3 5x + 1 ≤ -44 -2x + 15 < 7 (x + 15) / 2 < -5 (x + 9) / -3 > 3 (x - 6) / -10 ≥ -4 -5x / 4 ≥ 10 3x / 7 < 1 -4x / 3 > -12 x / 6 + 14 > 9 x / -7 + 5 ≤ 12 x / 2 - 10 < -19 2x - 7x ≤ 3 -3(x - 5) > 21 9 - x ≤ 7

Explanation:

Step1: Isolate the variable term

For each inequality, we first perform operations like addition, subtraction to get the term with the variable on one side. For example, in \(5x + 1\leq - 44\), we subtract 1 from both sides to get \(5x\leq-45\).

Step2: Solve for the variable

After isolating the variable term, we perform multiplication or division operations. When multiplying or dividing by a negative number, we reverse the inequality sign according to the rules of solving inequalities. For example, in \(-3x>17\), dividing by - 3 gives \(x<-\frac{17}{3}\)

Answer:

1. For \(-3x + 5>22\):

• First, subtract 5 from both sides: \(-3x>22 - 5\), so \(-3x>17\).
• Then divide both sides by - 3 and reverse the inequality sign: \(x<-\frac{17}{3}\)

1. For \(5x + 1\leq - 44\):

• Subtract 1 from both sides: \(5x\leq-44 - 1\), so \(5x\leq-45\).
• Divide both sides by 5: \(x\leq - 9\)

1. For \(-2x+15 < 7\):

• Subtract 15 from both sides: \(-2x<7 - 15\), so \(-2x<-8\).
• Divide both sides by - 2 and reverse the inequality sign: \(x > 4\)

1. For \(\frac{x + 15}{2}<-5\):

• Multiply both sides by 2: \(x + 15<-10\).
• Subtract 15 from both sides: \(x<-10 - 15\), so \(x<-25\)

1. For \(\frac{x + 9}{-3}>3\):

• Multiply both sides by - 3 and reverse the inequality sign: \(x + 9<-9\).
• Subtract 9 from both sides: \(x<-9 - 9\), so \(x<-18\)

1. For \(\frac{x - 6}{-10}\geq - 4\):

• Multiply both sides by - 10 and reverse the inequality sign: \(x - 6\leq40\).
• Add 6 to both sides: \(x\leq40 + 6\), so \(x\leq46\)

1. For \(-\frac{5x}{4}\geq10\):

• Multiply both sides by \(-\frac{4}{5}\) and reverse the inequality sign: \(x\leq10\times(-\frac{4}{5})\), so \(x\leq - 8\)

1. For \(\frac{3x}{7}<1\):

• Multiply both sides by 7: \(3x<7\).
• Divide both sides by 3: \(x<\frac{7}{3}\)

1. For \(-\frac{4x}{3}>-12\):

• Multiply both sides by \(-\frac{3}{4}\) and reverse the inequality sign: \(x < 9\)

1. For \(\frac{x}{6}+14>9\):

• Subtract 14 from both sides: \(\frac{x}{6}>9 - 14\), so \(\frac{x}{6}>-5\).
• Multiply both sides by 6: \(x>-30\)

1. For \(\frac{x}{-7}+5\leq12\):

• Subtract 5 from both sides: \(\frac{x}{-7}\leq12 - 5\), so \(\frac{x}{-7}\leq7\).
• Multiply both sides by - 7 and reverse the inequality sign: \(x\geq - 49\)

1. For \(\frac{x}{2}-10<-19\):

• Add 10 to both sides: \(\frac{x}{2}<-19 + 10\), so \(\frac{x}{2}<-9\).
• Multiply both sides by 2: \(x<-18\)

1. For \(2x-7x\leq3\):

• Combine like - terms: \(-5x\leq3\).
• Divide both sides by - 5 and reverse the inequality sign: \(x\geq-\frac{3}{5}\)

1. For \(-3(x - 5)>21\):

• First, distribute the - 3: \(-3x+15>21\).
• Subtract 15 from both sides: \(-3x>21 - 15\), so \(-3x>6\).
• Divide both sides by - 3 and reverse the inequality sign: \(x<-2\)

1. For \(9 - x\leq7\):

• Subtract 9 from both sides: \(-x\leq7 - 9\), so \(-x\leq - 2\).
• Multiply both sides by - 1 and reverse the inequality sign: \(x\geq2\)