Find a General Solution to Y Absolute Value
Solving Absolute Value Equations of the Type | x | = k .
Absolute value equations are useful in determining distance and error measurements.
The examples that we will consider are:
| x | = 3
| x – 6 | = 4
| 2 x – 3 | = 9
| x + 7 | = – 2
| x+ 8 | = | 3x – 4 |
Example 1 : Solve for x : | x | = 3
Solution.
This equation is asking us to find all numbers, x , that are 3 units from zero on the number line.
We must consider numbers both to the right and to the left of zero on the number line.
Notice that both 3 and -3 are three units from zero.
The solution is: x = 3 or x = −3 .
Example 1 suggests a rule that we can use when solving absolute value equations.
If c is a positive number, then | x | = c is equivalent to x = c or x = – c.
Example 2 : Solve for x : | x – 6 | = 4
Solution.
Step 1. Break the equation up into two equivalent equations using the rule: If | x | = c then x = c or x = - c .
| x – 6 | = 4 is equivalent to x – 6 = 4 or x – 6 = – 4
Step 2. Solve each equation .
x – 6 + 6 = 4 + 6
x = 10
x – 6 + 6 = – 4 + 6
x = 2
Step 3 . Check the solutions.
| 10 – 6 | = | 4 | = 4
| 2 – 6 | = | – 4 | = 4
The solutions are x = 10 and x = 2 .
Example 3 : Solve for x : | 2 x – 3 | = 9
Solution.
Step 1.
Break the equation up into two equivalent equations using the rule: If | x | = c then x = c or x = - c .
| 2 x – 3 | = 9 is equivalent to 2 x – 3 = 9 or 2 x – 3 = -9
Step 2. Solve each equation .
2 x – 3 = 9 or 2 x – 3 = -9
2 x – 3 + 3 = 9 + 3 or 2 x – 3 + 3 = -9 + 3
2 x = 12 or 2 x = -6
2 x ÷ 2 = 12 ÷ 2 or 2 x ÷ 2 = -6 ÷ 2
x = 6 or x = -3
Step 3 . Check the solutions.
x = 6: | 2(6) – 3 | = | 12 – 3 | = | 9 | = 9
x = -3: | 2(-3) – 3 | = | -6 – 3 | = | -9 | = 9
The solutions are x = 6 and x = -3 .
Example 4 : Solve for x : | x + 7 | = – 2
Solution.
The absolute value of a number is never negative. This equation has no solution .
Solving Absolute Value Equations of the Type | x | = | y |.
If the absolute values of two expressions are equal, then either the two expressions are equal, or they are opposites.
If x and y represent algebraic expressions, | x | = | y | is equivalent to x = y or x = – y.
Example 5 : Solve for x : | x + 8 | = | 3 x – 4 |
Solution.
Step 1. Break the equation up into two equivalent equations .
| x + 8 | = | 3 x – 4 | is equivalent to x + 8 = 3 x – 4 or x + 8 = – (3 x – 4)
Step 2. Solve each equation.
x + 8 = 3 x – 4 or x + 8 = – (3 x – 4)
x + 8 = 3 x – 4 or x + 8 = – 3 x + 4
x + 8 – x = 3 x – 4 – x or x + 8 + 3 x = -3 x + 4 + 3 x
8 = 2 x – 4 or 4 x + 8 = 4
8 + 4 = 2 x – 4 + 4 or 4 x + 8 – 8 = 4 – 8
12 = 2 x or 4 x = – 4
12 ÷ 2 = 2 x ÷ 2 or 4 x ÷ 4 = – 4 ÷ 4
6 = x or x = – 1
Step 3 . Check the solutions.
x = 6: | 6 + 8 | = | 3(6) – 4 |
| 14 | = | 18 – 4 |
| 14 | = | 14 |
14 = 14
x = – 1: | – 1 + 8 | = | 3( – 1) – 4 |
| 7 | = | – 3 – 4 |
| 7 | = | – 7 |
7 = – 7
The solutions are x = 6 and x = – 1 .
Find a General Solution to Y Absolute Value
Source: https://sites.austincc.edu/tsiprep/math-review/more-equations/solving-absolute-value-equations/