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After studying this lesson, you will be able to:

  • Solve equations containing fractions.

To Solve Equations containing fractions:

Clear out the fractions by multiplying the ENTIRE equation by the common denominator.

 

SPECIAL CASES

Sometimes when dealing with equations, all the variable cancel out. When this happens we have a "special case". If the expression we're left with is a true mathematical statement, our solution is called the Identity Property which means that there is an infinite number of solutions. The symbol for infinity is . If the expression we're left with is a false mathematical statement, there will be no solution. We indicate this with the empty set symbol which is Ø.

 

Example 1

2x + 5 = 2x -3 To solve this equation, we need to attempt to get the variables together on the same side.
2x + 5 - 2x = 2x -3 - 2x When we subtract 2x from each side, all variables are eliminated, leaving us with 5 = -3
5 = -3 We know this is a "special case" since we have no variables. Since 5 is not equal to -3, this is a false statement. Therefore there are no solutions.
Ø We write the empty set symbol for our answer

 

Example 2

3 ( x + 1 ) - 5 = 3x -2 To solve this equation, we first remove the parentheses by distributing.
3x + 3 - 5 = 3x - 2 We need to collect like terms (3 - 5 ) on the left side
3x - 2 = 3x -2 To solve this equation, we need to attempt to get the variables together on the same side.
3x - 2 - 3x = 3x -2 - 3x We do this by subtracting 3x from each side.

This gives us -2 = -2

-2 = -2 We know this is a "special case" since we have no variables. Since -2 is equal to -32, this is a true statement (the identity property). Therefore there are infinite number of solutions.
We write the symbol for infinity for our answer