Solving Absolute Value Equations

• Absolute value represents the distance of a number from zero on the number line.
• The absolute value of a number ( x ) is denoted as ( |x| ).
• ( |x| = x ) if ( x \geq 0 ) and ( |x| = -x ) if ( x < 0 ).
• Absolute value is never negative.

Solving Absolute Value Equations

• To solve an absolute value equation of the form ( |ax + b| = c ), where ( c \geq 0 ), set up two separate equations: ( ax + b = c ) and ( ax + b = -c ).
• Solve each equation separately to find all possible solutions for ( x ).
• Check each solution by substituting it back into the original absolute value equation to ensure it holds true.
• Extraneous solutions can occur, so checking is crucial.
• If ( c < 0 ), the absolute value equation ( |ax + b| = c ) has no solution, because absolute value cannot be negative.

Example: Solving ( |2x - 1| = 5 )

• Set up two equations: ( 2x - 1 = 5 ) and ( 2x - 1 = -5 ).
• Solve ( 2x - 1 = 5 ):
  • Add 1 to both sides: ( 2x = 6 ).
  • Divide by 2: ( x = 3 ).
• Solve ( 2x - 1 = -5 ):
  • Add 1 to both sides: ( 2x = -4 ).
  • Divide by 2: ( x = -2 ).
• Check ( x = 3 ): ( |2(3) - 1| = |6 - 1| = |5| = 5 ) (Correct).
• Check ( x = -2 ): ( |2(-2) - 1| = |-4 - 1| = |-5| = 5 ) (Correct).
• The solutions are ( x = 3 ) and ( x = -2 ).

Solving Absolute Value Inequalities

• For ( |ax + b| < c ), where ( c > 0 ), rewrite the inequality as a compound inequality: ( -c < ax + b < c ).
• Solve the compound inequality by isolating ( x ) in the middle.
• For ( |ax + b| > c ), where ( c > 0 ), rewrite the inequality as two separate inequalities: ( ax + b > c ) or ( ax + b < -c ).
• Solve each inequality separately.
• If ( c \leq 0 ) and the inequality is ( |ax + b| < c ), there is no solution.
• If ( c < 0 ) and the inequality is ( |ax + b| > c ), all real numbers are solutions.

Solving ( |ax + b| \leq c )

• Rewrite as ( -c \leq ax + b \leq c ).
• Isolate ( x ) to find the solution interval.

Solving ( |ax + b| \geq c )

• Rewrite as ( ax + b \geq c ) or ( ax + b \leq -c ).
• Solve each inequality separately to find the solution intervals.

Example: Solving ( |3x + 2| < 4 )

• Rewrite as ( -4 < 3x + 2 < 4 ).
• Subtract 2 from all parts: ( -6 < 3x < 2 ).
• Divide by 3: ( -2 < x < \frac{2}{3} ).
• The solution is the interval ( \left(-2, \frac{2}{3}\right) ).

Example: Solving ( |2x - 1| \geq 3 )

• Rewrite as ( 2x - 1 \geq 3 ) or ( 2x - 1 \leq -3 ).
• Solve ( 2x - 1 \geq 3 ):
  • Add 1 to both sides: ( 2x \geq 4 ).
  • Divide by 2: ( x \geq 2 ).
• Solve ( 2x - 1 \leq -3 ):
  • Add 1 to both sides: ( 2x \leq -2 ).
  • Divide by 2: ( x \leq -1 ).
• The solution is ( x \leq -1 ) or ( x \geq 2 ).

Key Considerations

• When multiplying or dividing by a negative number in an inequality, remember to reverse the inequality sign.
• Always check solutions by substituting them back into the original inequality to ensure they hold true.
• Pay close attention to whether the inequality is strict (( < ) or ( > )) or includes equality (( \leq ) or ( \geq )) because this affects whether the endpoints are included in the solution.
• The absolute value |x| is always non-negative, meaning ( |x| \geq 0 ) for all x.