Solving Linear Inequalities in Algebra

Solving Linear Inequalities

Rules for Solving Linear Inequalities

• When adding or subtracting the same value to both sides of an inequality, the direction of the inequality remains the same.
• When multiplying or dividing both sides of an inequality by a positive value, the direction of the inequality remains the same.
• When multiplying or dividing both sides of an inequality by a negative value, the direction of the inequality is reversed.

Steps for Solving Linear Inequalities

1. Simplify the inequality by combining like terms and removing any parentheses or fractions.
2. Isolate the variable by adding or subtracting the same value to both sides of the inequality.
3. Check the direction of the inequality and flip it if necessary when multiplying or dividing by a negative value.
4. Simplify the solution by writing it in its simplest form.

Examples of Solving Linear Inequalities

• Simple Inequality: 2x + 3 > 5
  • Subtract 3 from both sides: 2x > 2
  • Divide both sides by 2: x > 1
• Compound Inequality: -3x + 2 ≤ 5 and x - 2 > -3
  • Solve each inequality separately:
    • -3x + 2 ≤ 5 => -3x ≤ 3 => x ≥ -1
    • x - 2 > -3 => x > -1
  • Combine the solutions: x ≥ -1 and x > -1 => x > -1
• Inequality with Fractions: (x + 1)/2 ≥ 3
  • Multiply both sides by 2: x + 1 ≥ 6
  • Subtract 1 from both sides: x ≥ 5

Solving Linear Inequalities

Rules for Solving Linear Inequalities

• Adding or subtracting the same value to both sides of an inequality does not change the direction of the inequality.
• Multiplying or dividing both sides of an inequality by a positive value does not change the direction of the inequality.
• Multiplying or dividing both sides of an inequality by a negative value reverses the direction of the inequality.

Steps for Solving Linear Inequalities

• Simplify the inequality by combining like terms and removing any parentheses or fractions.
• Isolate the variable by adding or subtracting the same value to both sides of the inequality.
• Check the direction of the inequality and flip it if necessary when multiplying or dividing by a negative value.
• Simplify the solution by writing it in its simplest form.

Examples of Solving Linear Inequalities

• To solve a simple inequality, such as 2x + 3 > 5, subtract 3 from both sides and then divide both sides by 2 to get x > 1.
• To solve a compound inequality, such as -3x + 2 ≤ 5 and x - 2 > -3, solve each inequality separately and then combine the solutions.
• To solve an inequality with fractions, such as (x + 1)/2 ≥ 3, multiply both sides by 2 and then subtract 1 from both sides to get x ≥ 5.

Learn the rules and steps for solving linear inequalities, including how to maintain or reverse the direction of the inequality when performing operations.