Solving Inequalities

Solving and graphing inequalities is an essential skill in algebra and is used to represent relationships between variables where one value is greater than or less than another. Inequalities are often solved and graphed in a manner similar to equations, but there are some key differences.

Solving Inequalities:

• Inequality Symbols:

  • There are several inequality symbols, including:

    • < (less than)

    • > (greater than)

    • ≤ (less than or equal to)

    • ≥ (greater than or equal to)

    • ≠ (not equal to)

• Solving Linear Inequalities: To solve linear inequalities, use the same principles as solving linear equations, with one exception. When you multiply or divide both sides by a negative number, reverse the inequality symbol.

Example 1: Solve for x in the inequality "3x - 2 < 7."

• • Add 2 to both sides: 3x - 2 + 2 < 7 + 2.

  • Simplify: 3x < 9.

  • Divide both sides by 3 (positive, so the inequality symbol remains the same): (3x) / 3 < 9 / 3.

  • Simplify: x < 3.

Example 2: Solve for y in the inequality "-2y ≥ 10."

• • Divide both sides by -2 (negative, so reverse the inequality symbol): (-2y) / -2 ≤ 10 / -2.

  • Simplify: y ≤ -5.

Graphing Inequalities:

• Number Line: To graph inequalities, you can use a number line. On the number line, use an open circle for "<" or ">", a closed circle for "≤" or "≥," and shade the region that represents the solution.
  Example 3: Graph the solution to the inequality "x ≤ 4."

  • Use a closed circle at 4 (because of "≤") and shade to the left.

  • The graph represents all values less than or equal to 4.

Example 4: Graph the solution to the inequality "y > 2."

• • Use an open circle at 2 (because of ">") and shade to the right.

  • The graph represents all values greater than 2.

• Compound Inequalities: Compound inequalities involve multiple inequalities connected by "and" (⋀) or "or" (⋁). You can graph the solutions by combining the graphs of individual inequalities.
  Example 5: Graph the solution to the compound inequality "2 < x ≤ 5."

  • Use an open circle at 2 and a closed circle at 5.

  • Shade between the two circles.

  • The graph represents all values greater than 2 and less than or equal to 5.

Solving and graphing inequalities is a valuable skill for modeling real-world situations, such as inequalities involving income, age, temperature, and more. By understanding and applying these concepts, you can make informed decisions and analyze data in various fields.