There are several ways to calculate x, whether you are working with exponents and radicals or you have to divide or multiply. No matter what process you use, you should always find a way to isolate x to one side of the equation to find its value. Read on to find out how to do it.
Steps
Part 1 of 5: Using a Basic Linear Equation
Step 1. Write the problem
Here it is:
- 22(x + 3) + 9 - 5 = 32
Step 2. Solve the exponent
Remember the order of operations: PEMDAS (parentheses, exponents, multiplication or division, and addition or subtraction). You can't solve the parentheses first because x is inside the parentheses, so you must start with the exponent, 22. 22 = 4
4 (x + 3) + 9 - 5 = 32
Step 3. Do the multiplication
Just distribute the 4 in (x + 3). In this way:
4x + 12 + 9 - 5 = 32
Step 4. Do the addition and subtraction
Just add or subtract the rest of the numbers. In this way:
- 4x + 21-5 = 32
- 4x + 16 = 32
- 4x + 16 - 16 = 32 - 16
- 4x = 16
Step 5. Isolate the variable
To do this, just divide both sides of the equation by 4 to find x. 4x / 4 = x and 16/4 = 4, so x = 4.
- 4x / 4 = 16/4
- x = 4
Step 6. Check your work
Just substitute 4 for x in the original equation to make sure it's okay. In this way:
- 22(x + 3) + 9 - 5 = 32
- 22(4+3)+ 9 - 5 = 32
- 22(7) + 9 - 5 = 32
- 4(7) + 9 - 5 = 32
- 28 + 9 - 5 = 32
- 37 - 5 = 32
- 32 = 32
Part 2 of 5: Using Exponents
Step 1. Write the problem
If you are working with a problem in which the x term includes an exponent:
- 2x2 + 12 = 44
Step 2. Isolate the term with the exponent
The first thing to do is combine the like terms in such a way that all the constant terms are on the right side of the equation while the term with the exponent is on the left side. Just subtract 12 from both sides. In this way:
- 2x2+12-12 = 44-12
- 2x2 = 32
Step 3. Isolate the variable with the exponent by dividing both sides by the coefficient of the x term
In this case, 2 is the coefficient of x, so divide both sides of the equation by 2 to get rid of the coefficient. In this way:
- (2x2)/2 = 32/2
- x2 = 16
Step 4. Take the square root of each side of the equation
Take the square root of x2 will cancel it. So take the square root of both sides. You will have an x on one side and the square root of 16, 4, on the other. Therefore, x = 4.
Step 5. Check your work
Just substitute 4 for x in the original equation to make sure it's okay. In this way:
- 2x2 + 12 = 44
- 2 x (4)2 + 12 = 44
- 2 x 16 + 12 = 44
- 32 + 12 = 44
- 44 = 44
Part 3 of 5: Using Fractions
Step 1. Write the problem
Let's say you work with the following problem:
(x + 3) / 6 = 2/3
Step 2. Do the rule of three
To do the rule of three, just multiply the denominator of each fraction by the numerator of the other fraction (essentially, multiply on two diagonal lines). So, multiply the first denominator (6) by the second numerator (2) to get 12 on the right side of the equation. Multiply the second denominator (3) by the first numerator (x + 3) to get 3x + 9 on the left side of the equation. It should look like this:
- (x + 3) / 6 = 2/3
- 6 x 2 = 12
- (x + 3) x 3 = 3x + 9
- 3x + 9 = 12
Step 3. Combine like terms
Combine the constant terms in the equation to subtract 9 from both sides. This is what you should do:
- 3x + 9 - 9 = 12 - 9
- 3x = 3
Step 4. Isolate x by dividing each term by the coefficient of x
Just divide 3x and 9 by 3 (the coefficient of x) to find x. 3x / 3 = x and 3/3 = 1, so you will get x = 1.
Step 5. Check your work
To check your work, just plug x into the original equation to make sure it works. This is what you should do:
- (x + 3) / 6 = 2/3
- (1 + 3)/6 = 2/3
- 4/6 = 2/3
- 2/3 = 2/3
Part 4 of 5: Using Radical Signs
Step 1. Write the problem
Let's say you want to calculate x in the following problem:
√ (2x + 9) - 5 = 0
Step 2. Isolate the square root
You must move the part of the equation with the square root sign to the side before you can proceed. This way, you will have to add 5 to both sides of the equation. You should do it this way:
- √ (2x + 9) - 5 + 5 = 0 + 5
- √ (2x + 9) = 5
Step 3. Square both sides
Just as you divide both sides of an equation by a coefficient that is multiplied by x, you must square both sides of the equation if x appears under the square root or the radical sign. This will remove the radical sign from the equation. You must do it this way:
- (√ (2x + 9))2 = 52
- 2x + 9 = 25
Step 4. Combine like terms
Combine like terms by subtracting 9 from both sides so that all constant terms are on the right side of the equation, while x must remain on the left side. This is what you should do:
- 2x + 9 - 9 = 25 - 9
- 2x = 16
Step 5. Isolate the variable
The last thing you need to do to find the value of x is to isolate the variable by dividing both sides of the equation by 2 (the coefficient of the x term). 2x / 2 = x and 16/2 = 8, this way you will get x = 8.
Step 6. Check your work
Substitute 8 for x in the original equation to get the correct answer:
- √ (2x + 9) - 5 = 0
- √(2(8)+9) - 5 = 0
- √(16+9) - 5 = 0
- √(25) - 5 = 0
- 5 - 5 = 0
Part 5 of 5: Use an absolute value
Step 1. Write the problem
Let's say you try to calculate x in the following problem:
| 4x +2 | - 6 = 8
Step 2. Isolate the absolute value
The first thing to do is combine the like terms and get the terms inside the absolute value sign to one side. In this case, you should do it by adding 6 to both sides of the equation. In this way:
- | 4x +2 | - 6 = 8
- | 4x +2 | - 6 + 6 = 8 + 6
- | 4x +2 | = 14
Step 3. Eliminate the absolute value and solve the equation
This is the first and easiest step. You will have to calculate the x twice when working with an absolute value. You should do it this way the first time:
- 4x + 2 = 14
- 4x + 2 - 2 = 14 -2
- 4x = 12
- x = 3
Step 4. Eliminate the absolute value and change the sign of the terms on the opposite side of the equal sign before solving it
Now do it again, only this time set the first part of the equation equal to -14 instead of 14. Like this:
- 4x + 2 = -14
- 4x + 2 - 2 = -14 - 2
- 4x = -16
- 4x / 4 = -16/4
- x = -4
Step 5. Check your work
Now that you know that x = (3, -4), just plug both numbers into the original equation to see if it works. In this way:
-
(For x = 3):
- | 4x +2 | - 6 = 8
- |4(3) +2| - 6 = 8
- |12 +2| - 6 = 8
- |14| - 6 = 8
- 14 - 6 = 8
- 8 = 8
-
(For x = -4):
- | 4x +2 | - 6 = 8
- |4(-4) +2| - 6 = 8
- |-16 +2| - 6 = 8
- |-14| - 6 = 8
- 14 - 6 = 8
- 8 = 8
Advice
- Radicals or roots are another way to represent exponents. The square root of x = x ^ 1/2.
- To check your work, plug the value of x into the original equation and solve it.