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Question:

Solve the following equation for x.
|14x – 1⁄3| = 2

A 1⁄6 , –5⁄42
Explaination

When solving an equation with one variable, your goal is to isolate the variable on one side of the equal sign. You do this by moving all of the other numbers to the opposite of the equal sign. You move numbers by performing the opposite operation (subtraction/addition and multiplication/division), on both sides of the equation to keep it balanced.
|14x – 1⁄3| = 2
In this equation, we see the use of the absolute value symbol, | |.
The absolute value of a number is the magnitude of that number. In other words, it is the number of places it is from zero. This means that the absolute value of a negative number is simply the number of spaces it is away from zero (in either direction). For example, |- 3| = 3 because -3 is three places from zero.
When there is an absolute value within an equation, it represents the absolute value of that number, when x is known.
When we see the absolute value symbol, it alerts us that the value of the number inside can be either positive or negative.
Therefore, we must solve for both possibilities, and separate the problem into the positive answer and the negative answer and solve for both:
Situation 1: One version of the equation should represent what is inside the absolute value symbols as a positive number. Remember that it is the entire 14x – 1/3 that could be a positive number here.
14x – 1⁄3 = 2
Add 1⁄3 to each side to eliminate the (-1⁄3 ) from the left-hand side of the equation.
14x – 1⁄3 + 1⁄3 = 2 + 1⁄3
14x = 2 1⁄3
Now, you need to divide both sides of the equation by the coefficient (14) to isolate the variable completely. Convert the mixed number to an improper fraction in order to divide the fractions.
2 1⁄3 → (2 • 3) + 1 = 7 → 7⁄3
14x = 7⁄3
14x ÷ 14= 7⁄3 ÷ 14
Remember that when dividing fractions, you multiply by the reciprocal (flip the second fraction and multiply). Since 14 is a whole number, you can make it a fraction by making 14 the numerator and 1 the denominator (14⁄1). Then, flip it (1⁄14) and multiply the two fractions.
x = 7⁄3 × 1⁄14
x = 7⁄42
Reduce by finding a common factor between the two (7) and dividing both the numerator and denominator by that number.
x = 1⁄6
Situation 2:One version of the equation should represent what is inside the absolute value symbols as a positive number. Remember that it is the entire 14x – 1/3 that could be a positive number here. -(14x – 1⁄3) = 2
Distribute the negative sign to all values included in the absolute value of the original equation.
-14x + 1⁄3 = 2
Subtract ⅓ from each side of the equation in order to eliminate the ⅓ from the left-hand side of the equation.
-14x + 1⁄3– 1⁄3 = 2 – 1⁄3
-14x = 2 – 1⁄3
Determine a common denominator in order to subtract the fractions. Remember that whole numbers are written as fractions by making the whole number the numerator and 1 the denominator (2⁄1).
-14x = 2⁄1 – 1⁄3
The lowest common denominator for 1 and 3 is 3. In order to make the denominator of 2⁄1 a 3, you need to multiply both the numerator and the denominator by 3 (because 1 • 3 = 3).
-14x = 6⁄3 – 1⁄3
-14x = 5⁄3
Now, you need to divide both sides of the equation by the coefficient (14) to isolate the variable completely. Remember that when dividing fractions, you multiply by the reciprocal (flip the second fraction and multiply). Since 14 is a whole number, you can make it a fraction by making 14 the numerator and 1 the denominator (14⁄1). Then, flip it (1⁄14) and multiply the two fractions.
-14x ÷ -14= 5⁄3 ÷ –14⁄1
x = 5⁄3 × –1⁄14
x = –5⁄42