3. Solving Absolute Value Equations
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The first step is isolating the absolute value expression.
See solution.
There are two main steps when solving an absolute value equation.
| Step 1 | Isolate the absolute value expression. |
|---|---|
| Step 2 | If the absolute value expression is equal to a positive number, solve both arising equations from the disjunction (positive and negative). There are two solutions. |
| If the absolute value expression is equal to zero, remove the absolute value and solve the equation. There is one solution. | |
| If the absolute value expression is equal to a negative number, there is no solution. |
Let's see examples for these cases.
Let's consider an equation.
2|2x-1|+3=13
Now, since the absolute value expression is equal to a positive number, we will solve both arising equations from the disjunction.
lc 2x-1 ≥ 0:2x-1 = 5 & (I) 2x-1 < 0:2x-1 = - 5 & (II)
(I), (II): LHS+1=RHS+1
(I), (II): .LHS /2.=.RHS /2.
We found two solutions.
Consider now a second equation. 2|x+5|+3=3 Again, the first step is isolating the absolute value expression.
This time, the absolute value expression is equal to zero. Therefore, we will remove the absolute value and solve the equation.
We found one solution.
Finally, consider one last equation. 5|3x-10|+6=1 As we did before, we will start by isolating the absolute value expression.
Since the absolute value expression is equal to a negative number, the equation has no solutions.