how do you solve |4x-5|=|2x+3|
4x-5=2x+3
You need to get all the variables (letters, which would be x in this equation) on one side and the numbers on the other.
Adding 5 to both sides:
4x=2x+8
Subtracting 2x from both sides
2x=8
Dividing both sides by 2
x=?
That correctly provides one possible solution out of two.
Because of the absolute value function, the equation can also be solved as:
(4x-5)=-(2x+3)
which can be transformed to
6x=2, or x=1/3
To solve an equation or an inequality involving absolute value, you need to consider both the positive and negative cases separately. Let's break down the process step by step to solve the equation |4x - 5| = |2x + 3|:
Step 1: Set up the positive case
In the positive case, you remove the absolute value bars and consider the expression inside the bars as it is. Therefore, the equation becomes:
4x - 5 = 2x + 3
Step 2: Solve the positive case equation
Now, solve the positive case equation as you would with any linear equation:
4x - 5 = 2x + 3
Subtract 2x from both sides:
4x - 2x - 5 = 2x - 2x + 3
This simplifies to:
2x - 5 = 3
Add 5 to both sides:
2x - 5 + 5 = 3 + 5
This simplifies to:
2x = 8
Divide both sides by 2 to isolate x:
(2x)/2 = 8/2
This simplifies to:
x = 4
So, for the positive case, x = 4.
Step 3: Set up the negative case
Now, consider the negative case. In this case, you change the signs of the expressions inside the absolute value bars. Therefore, the equation becomes:
4x - 5 = -(2x + 3)
Step 4: Solve the negative case equation
Now, solve the negative case equation just like any other linear equation:
4x - 5 = -(2x + 3)
Distribute the negative sign:
4x - 5 = -2x - 3
Add 2x to both sides:
4x + 2x - 5 = -2x + 2x - 3
This simplifies to:
6x - 5 = -3
Add 5 to both sides:
6x - 5 + 5 = -3 + 5
This simplifies to:
6x = 2
Divide both sides by 6 to isolate x:
(6x)/6 = 2/6
This simplifies to:
x = 1/3
So, for the negative case, x = 1/3.
Step 5: Check the solutions
It's important to check if the potential solutions satisfy the original equation. Substitute the values of x back into the original equation and verify that both sides are equal.
In our case, substituting x = 4 into the original equation:
|4x - 5| = |2x + 3|
|4(4) - 5| = |2(4) + 3|
|16 - 5| = |8 + 3|
|11| = |11|
Both sides of the equation are equal when x = 4.
Similarly, substituting x = 1/3 into the original equation:
|4x - 5| = |2x + 3|
|4(1/3) - 5| = |2(1/3) + 3|
|-4/3 - 5| = |2/3 + 3|
|-(4/3 + 15/3)| = |2/3 + 9/3|
|-(19/3)| = |11/3|
Both sides of the equation are equal when x = 1/3.
Therefore, the solution to the equation |4x - 5| = |2x + 3| is x = 4 and x = 1/3.