Solving Linear Inequalities

Solving Linear Inequalities

Graphical Method

• Graph the related linear equation on the number line
• Shade the region that satisfies the inequality:
  • Left side of the line for < or ≤
  • Right side of the line for > or ≥
• The solution is the shaded region

Algebraic Method

• Isolate the variable on one side of the inequality
• Flip the inequality sign when multiplying or dividing by a negative number
• Simplify the inequality to find the solution

Properties of Inequality

• Addition and Subtraction Property: Adding or subtracting the same value to both sides of an inequality does not change the solution
• Multiplication and Division Property: Multiplying or dividing both sides of an inequality by a positive number does not change the solution; multiplying or dividing by a negative number reverses the inequality sign

Compound Inequalities

• And (Intersection):
  • Solve each inequality separately
  • Find the intersection of the solutions
• Or (Union):
  • Solve each inequality separately
  • Find the union of the solutions

Solving Systems of Linear Inequalities

• Graph the related linear equations on the same coordinate plane
• Shade the region that satisfies all the inequalities
• The solution is the shaded region

Solving Linear Inequalities

Graphical Method

• To solve linear inequalities graphically, graph the related linear equation on the number line.
• Shade the region that satisfies the inequality, considering the direction of the inequality sign:
  • Shade to the left for ≤ or ≥.
• The solution to the inequality is the shaded region.

Algebraic Method

• To solve linear inequalities algebraically, isolate the variable on one side of the inequality.
• When multiplying or dividing by a negative number, flip the inequality sign.
• Simplify the inequality to find the solution.

Properties of Inequalities

• The Addition and Subtraction Property states that adding or subtracting the same value to both sides of an inequality does not change the solution.
• The Multiplication and Division Property states that:
  • Multiplying or dividing both sides of an inequality by a positive number does not change the solution.
  • Multiplying or dividing by a negative number reverses the inequality sign.

Compound Inequalities

• To solve compound inequalities with "and" (intersection), solve each inequality separately and find the intersection of the solutions.
• To solve compound inequalities with "or" (union), solve each inequality separately and find the union of the solutions.

Solving Systems of Linear Inequalities

• Graph the related linear equations on the same coordinate plane.
• Shade the region that satisfies all the inequalities.
• The solution to the system of linear inequalities is the shaded region.