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}
May 19, 19 09:20 AM
Basic math review game - The one and only math adventure game online. Embark on a quest to solve math problems!
Basic math formulas
Algebra word problems
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.
Recommended
Scientific Notation Quiz
Graphing Slope Quiz
Adding and Subtracting Matrices Quiz
Factoring Trinomials Quiz
Solving Absolute Value Equations Quiz
Order of Operations Quiz
Types of angles quiz
May 19, 19 09:20 AM
Basic math review game - The one and only math adventure game online. Embark on a quest to solve math problems!