Solving Absolute Value Equations What is Absolute Value
Solving Absolute Value Equations
What is Absolute Value? • • • The absolute value of a number is the number of units it is from zero on the number line. 5 and -5 have the same absolute value. The symbol |x| represents the absolute value of the number x.
• • • |-8| = 8 |4| = 4 You try: • • • |15| = ? |-23| = ? Absolute Value can also be defined as : • • if a >0, then |a| = a if a < 0, then |a| = -a
We can evaluate expressions that contain absolute value symbols. • • Think of the | | bars as grouping symbols. Evaluate |9 x -3| + 5 if x = -2 |9(-2) -3| + 5 |-18 -3| + 5 |-21| + 5 21+ 5=26
Equations may also contain absolute value expressions • • When solving an equation, isolate the absolute value expression first. Rewrite the equation as two separate equations. • Consider the equation | x | = 3. The equation has two solutions since x can equal 3 or -3. Solve each equation. Always check your solutions. Example: Solve |x + 8| = 3 x + 8 = 3 and x + 8 = -3 x = -5 x = -11 Check: |x + 8| = 3 |-5 + 8| = 3 |3| = 3 3=3 |-11 + 8| = 3 |-3| = 3 3=3
Now Try These • Solve |y + 4| - 3 = 0 |y + 4| = 3 You must first isolate the variable by adding 3 to both sides. • • Write the two separate equations. y+4=3 & y + 4 = -3 y = -1 y = -7 Check: |y + 4| - 3 = 0 |-1 + 4| -3 = 0 |-7 + 4| - 3 = 0 |-3| - 3 = 0 3 -3=0 0=0
• |3 d - 9| + 6 = 0 First isolate the variable by subtracting 6 from both sides. |3 d - 9| = -6 There is no need to go any further with this problem! • • Absolute value is never negative. Therefore, the solution is the empty set!
• • Solve: 3|x - 5| = 12 |x - 5| = 4 x-5=4 and x=9 Check: 3|x - 5| = 12 3|9 - 5| = 12 3|4| = 12 3(4) = 12 12 = 12 x - 5 = -4 x=1 3|1 - 5| = 12 3|-4| = 12 3(4) = 12 12 = 12
Solve: |8 + 5 a| = 14 - a 8 + 5 a = 14 - a 6 a = 6 a=1 and 8 + 5 a = -(14 – a) Set up your 2 equations, but make sure to negate the entire right side of the second equation. and 8 + 5 a = -14 + a 4 a = -22 a = -5. 5 Check: |8 + 5 a| = 14 - a |8 + 5(1)| = 14 - 1 |13| = 13 13 = 13 |8 + 5(-5. 5) = 14 - (-5. 5) |-19. 5| = 19. 5
- Slides: 9