1.6 Absolute Value Equations and Inequalities

Absolute Value must be by itself on the left side. Absolute value: is a number’s distance from zero on the number line and it is nonnegative If x  0, then  x  = x or If x  0, then  x  = -x Distance cannot be negative!!! 8 = 8 – 8 = 8 8 8 Note: Absolute Value must be by itself on the left side.

Solving Absolute Value Equations This has two solutions, since it can be equal to – 6 or 6, because  – 6 = 6 and 6 = 6. 1. Solve 15 – 3x = 6. 15 – 3x = 6 or –15 –15 – 3x = – 9 – 3 – 3 x = 3 Check: 15 – 3x = 6 15 – 3(3) = 6 15 – 3(7) = 6 6 = 6 –6 = 6 15 – 3x = – 6 –15 –15 –3x = –21 – 3 – 3 or x = 7 {3, 7}

Extraneous Solutions -5 Extraneous Solutions: a solution of an equation derived from an original equation that is not a solution of the original equation 2. Solve 3x – 4 = -4x – 1 3x – 4 = -4x – 1 or 3x – 4 = –(-4x – 1) 7x = 3 x = 3/7 3(3/7) – 4 = -4(3/7) – 1 or 3(-5) – 4 = -4(-5) – 1 -15 – 4 = 20 – 1 19 = 19; true, solution or 3x – 4 = 4x + 1 or –5 = x √: (9/7) – 4 = (-12/7) – 1 19/7 = –19/7; false, extraneous solution -5

Absolute Value must be by itself on the left side. Absolute Value Inequalities Let k represent a positive real number. x > k is equivalent to x < -k or x > k. x  k is equivalent to -k  x  k. Outsides Insides Note: Absolute Value must be by itself on the left side.

Solving Inequalities of the Form A  b 3. Solve -2x + 1 + 5 ≥ -3. Graph the solution. - 5 -5 -2x + 1 ≥ -8 -2 -2 x + 1  4 -4  x + 1  4 -1 -1 -1 -5  x  3 Flip sign when you divide by a negative Graph

Solving Inequalities of the Form A > b 4. Solve 2x - 5  3. Graph the solution. 2x – 5 < -3 or 2x – 5  3 + 5 <+ 5 + 5 >+ 5 2x < 2 2x > 8 2 2 2 2 x < 1 or x > 4 Graph

True All Real numbers Distance cannot be negative!!! Translation: “The distance from zero on the number line is greater than -3”. Distance cannot be negative!!! The smallest distance would be zero. True All Real numbers

False No Solution No Graph Distance cannot be negative!!! Translation: “The distance from zero on the number line is less than -3”. Distance cannot be negative!!! The smallest distance would be zero. False No Solution No Graph

Solve and graph. White Boards