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Solving Absolute Value Equations
Distance is the amount of space between objects this value will always be positive. Because the absolute value is the length of this distance from zero on a number line; the absolute value of any number must be positive as well.
Definition of absolute value is the distance a number is from zero on a number line.
For example if we wanted to find out which number(s) have an absolute value of 5.
We want to check to see which numbers are 5 units from zero the number line.
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Negative 5 is a distance of 5 units going to the left
10 2 3 4 5 6 7 8 9 1012345678910
Positive 5 is a distance of 5 units going to the right.
For any positive number absolute value or the distance from zero will always have two values.
and
Using absolute value notation we can write :
The absolute vale of any real positive number will always be a positive number.
The absolute vale of any real negative number will always be a positive number.
EXAMPLE 1
SOLUTION
Solve the equation : , and graph the
solution(s) on a number line.
Let's use a number line to arrive at the solution.
Recall that the absolute value of a number is the distance a number is from zero on a number line.
Starting at zero if we move to the LEFT 4 units you end up at 4.
10 2 3 4 5 6 7 8 9 1012345678910
Starting at zero if we move to the RIGHT 4 units you end up at 4.
10 2 3 4 5 6 7 8 9 1012345678910
For this equation I need to find any number that is distance of 4 units from zero.
For any positive number the absolute value(the distance from zero) will always have two values.
Let's verify these solutions, substituting each of the values into the original equation and simplifying.
Change x to 4 Change x to 4
Both 4 and 4 satisfied the equation are
solution set is both numbers.
10 2 3 4 5 6 7 8 9 1012345678910
EXAMPLE 2
SOLUTION
Solve the equation : , and graph the solution(s) on a number line.
Let's use a number line to arrive at the solution.
Recall that the absolute value of a number is the distance a number is from zero on a number line.
For this equation I need to find any number that is distance of 4 units from zero.
If distance must always be a positive value, how can this be possible.
It is NOT possible to take the absolute value of any number and get a negative value.
NO SOLUTION
Let's use the equation to show why any equation like this will not have a solution.
10 2 3 4 5 6 7 8 9 1012345678910
We can try any numbers but the only numbers
that could possibly work are 4 & 4.
Let's see what happens :
Change x to 4 Change x to 4
Both equations are false, NO SOLUTION.
How to solve an absolute value equation without using a number line?
Step 1 :
Isolate the absolute value expression on one side of the equation.
Step 2 :
Eliminate the absolute value.
Step 4:
Verify the solution for each equation.
Step 3:
Solve each of the equations.
EXAMPLE 3
SOLUTION
Solve the equation : , and graph the solution(s) on a number line.
Step 1 :
Isolate the absolute value expression on one side of the equation.
Isolate
+11 +11
Now, that the absolute expression is isolated we can move onto Step 2.
Step 2 :
Eliminate the absolute value.
Take the expression inside of the absolute value bars and write 2 equations.
The first equation will be the expression inside of the absolute value bars equal to the number on the right side of the equation.
The second equation will be the expression inside of the absolute value bars equal to the OPPOSITE of the number on the right side of the equation.
** Never change the expression inside the absolute value when writing the 2 equations.
1st Equation 2nd Equation
Step 3:
Solve each equation.
+3 +3 +3 +3
Step 4:
Verify the solution for each equation.
Substitute each of the values back into the original equation and simplify.
Change x to 8 Change x to 2
Change x to 8 Change x to 2
Both 8 and 2 satisfied the equation are
solution set is both numbers.
Solution Set : 8, 2
10 2 3 4 5 6 7 8 9 1012345678910
EXAMPLE 4
SOLUTION
Solve the equation : , and graph the solution set on a number line.
Step 1 :
Isolate the absolute value expression on one side of the equation.
Isolate
Now, that the absolute expression is isolated we can move onto Step 2.
Step 2 :
Eliminate the absolute value.
Take the expression inside of the absolute value bars and write 2 equations.
The first equation will be the expression inside of the absolute value bars equal to the number on the right side of the equation.
The second equation will be the expression inside of the absolute value bars equal to the OPPOSITE of the number on the right side of the equation.
** Never change the expression inside the absolute value when get writing the 2 equations.
1st Equation 2nd Equation
Step 3:
Solve each of the equations.
+1 +1 +1 +1
Step 4:
Verify the solution for each equation.
Substitute each of the values back into the original equation and simplify.
Change x to 8 Change x to 7
Both 8 and 7 satisfied the equation are solution set is both numbers.
Solution Set : 7, 8
10 2 3 4 5 6 7 8 9 1012345678910
Change x to 8 Change x to 7
