Example #1: 6|2x + 3| - 7 = 2|2x + 3| + 1
Try to have the numbers on the right side. Add 7 to both sides of the equation.
6|2x + 3| - 7 + 7 = 2|2x + 3| + 1 + 7
6|2x + 3| = 2|2x + 3| + 8
Now, try to have the expressions 6|2x + 3| and 2|2x + 3| on the left side. Subtract 2|2x+3| from both sides of the equations.
6|2x + 3| - 2|2x + 3| = 2|2x + 3| - 2|2x + 3| + 8
6|2x + 3| - 2|2x + 3| = 8
Finally, you need to make an important observation.
Notice that 6|2x + 3| and -2|2x + 3| are like terms. Therefore, you can use the distributive property to simplify.
6|2x + 3| - 2|2x + 3| = (6 - 2)|2x + 3| = 4|2x + 3|
We end up with 4|2x + 3| = 8
Divide both sides of the equation by 4
(4/4)|2x + 3| = 8/4
1|2x + 3| = 2
|2x + 3| = 2
If 2x + 3 ≥0, 2x ≥ -3 and x ≥ -3/2 left -3/2 right
________________________________________
You can pick a number on the right of -3/2 or on the left of -3/2.
If you pick any number on the right of -3/2, 2x + 3 will be positive. For example, if you pick -1/2, 2(-1/2) + 3 = 2(-0.5) + 3 = -1 + 3 = 2.
Therefore, |2x + 3| = 2x + 3
You need to solve 2x + 3 = 2
2x + 3 = 2
2x + 3 - 3 = 2 - 3
2x = -1
x = -1/2
If you pick any number on the left of -3/2, 2x + 3 will be negative. For example, if you pick -2, 2(-2) + 3 = -4 + 3 = -1.
Therefore, |2x + 3| = -(2x + 3)
You need to solve -(2x + 3) = 2
-(2x + 3) = 2
2x + 3 = -2
2x + 3 - 3 = - 2 - 3
2x = -5
x = -5/2
Solution = {-5/2, -1/2}
Example #2: |2x + 6| + - 3 + |3x - 4| = 9
|2x + 6| + - 3 + 3 + |3x - 4| = 9 + 3
|2x + 6| + |3x - 4| = 12
This problem is complex because you have multiple cases to consider.
If 2x + 6 ≥0, 2x ≥ -6 and x ≥ -3The number that we pick could be on the left of -3, between -3 and 4/3, or on the right of 4/3.
Left -3 between 4/3 right _______________________________________________
On the left of -3, both 2x + 6 and 3x - 4 will be negative. For example, if x = -4
2x + 6 = 2 times -4 + 6 = -8 + 6 = -2
3x - 4 = 3 times -4 - 4 = -12 - 4 = -16
Therefore, |2x + 6| = -(2x + 6) and |3x - 4| = -(3x - 4)
We need to solve -(2x + 6) + -(3x - 4) = 12
Multiply everything by -1
-1 times -(2x + 6) + -1 times -(3x - 4) = -1 times 12
2x + 6 + 3x - 4 = -12
2x + 3x + 6 - 4 = -12
5x + 2 = -12
5x + 2 - 2 = -12 - 2
5x = -14
x = -14/5 = -2.8
This answer will make no sense since we chose our number on the left of -3. However, -2.8 is on the right of -3. So -2.8 will not solve the equation.
Between - 3 and 4/3, 2x + 6 is positive and 3x - 4 is negative. For example, if x = -1
2x + 6 = 2 times -1 + 6 = -2 + 6 = 4
3x - 4 = 3 times -1 - 4 = -3 - 4 = -7
Therefore, |2x + 6| = (2x + 6) and |3x - 4| = -(3x - 4)
We need to solve (2x + 6) + -(3x - 4) = 12
2x + 6 + -3x + 4 = 12
2x + -3x + 6 + 4 = 12
-x + 10 = 12
-x + 10 - 10 = 12 - 10
-x = 2
x = -2
-2 is between -3 and 4/3. Check the answer.
On the right of 4/3, both 2x + 6 and 3x - 4 will be positive. For example, if x = 3
2x + 6 = 2 times 3 + 6 = 6 + 6 = 12
3x - 4 = 3 times 3 - 4 = 9 - 4 = 5
Therefore, |2x + 6| = 2x + 6 and |3x - 4| = 3x - 4
We need to solve (2x + 6) + (3x - 4) = 12
2x + 6 + 3x - 4 = 12
2x + 3x + 6 - 4 = 12
5x + 2 = 12
5x + 2 - 2 = 12 - 2
5x = 10
x = 10/5 = 2
2 is on the right of 4/3. Check the answer.
Solution = { -2, 2}
Feb 02, 18 05:18 PM
These preschool math worksheets will help kids prepare for kindergarten math
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.
Feb 02, 18 05:18 PM
These preschool math worksheets will help kids prepare for kindergarten math
Our Top Pages
Formula for percentage
Compatible numbers
Basic math test
Basic math formulas
Types of angles
Math problem solver
Algebra word problems
Surface area of a cube
Finding the average
Scale drawings
Everything you need to prepare for an important exam!
K-12 tests, GED math test, basic math tests, geometry tests, algebra tests.
Real Life Math Skills
Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.