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.

Learn to solve linear inequalities using graphical and algebraic methods, understanding the properties of inequality signs and finding solutions.