Absolute Value Equations

Absolute Value Equations

Definition

• An absolute value equation is an equation that involves the absolute value of a variable or expression.
• The absolute value of a number is its distance from 0 on the number line.

Notation

• The absolute value of a number or expression is denoted by | |, for example: |x| or |2x - 3|.
• The absolute value can be thought of as the "distance" from 0, so |x| = a can be read as "x is a units away from 0".

Properties

• The absolute value of a number is always non-negative (or zero).
• The absolute value of a product is the product of the absolute values: |ab| = |a| |b|.
• The absolute value of a sum is not necessarily the sum of the absolute values: |a + b| ≠ |a| + |b|.

Solving Absolute Value Equations

• To solve an absolute value equation, isolate the absolute value expression on one side of the equation.
• Split the equation into two possible cases: one where the expression inside the absolute value is non-negative, and one where it is negative.
• Solve each case separately, and then combine the solutions.

Examples

• Solve the equation: |x| = 5
  • Case 1: x = 5
  • Case 2: x = -5
  • Solution: x = 5 or x = -5
• Solve the equation: |2x - 3| = 7
  • Case 1: 2x - 3 = 7
  • Case 2: 2x - 3 = -7
  • Solution: x = 5 or x = -2

Absolute Value Equations

• An absolute value equation involves the absolute value of a variable or expression, which is the distance from 0 on the number line.

Notation

• The absolute value of a number or expression is denoted by | |, such as |x| or |2x - 3|.
• The absolute value can be thought of as the "distance" from 0, so |x| = a can be read as "x is a units away from 0".

Properties

• The absolute value of a number is always non-negative (or zero).
• The absolute value of a product is the product of the absolute values: |ab| = |a| |b|.
• The absolute value of a sum is not necessarily the sum of the absolute values: |a + b| ≠ |a| + |b|.

Solving Absolute Value Equations

• To solve an absolute value equation, isolate the absolute value expression on one side of the equation.
• Split the equation into two possible cases: one where the expression inside the absolute value is non-negative, and one where it is negative.
• Solve each case separately, and then combine the solutions.

Examples

• The equation |x| = 5 has two solutions: x = 5 and x = -5.
• The equation |2x - 3| = 7 has two solutions: x = 5 and x = -2.