Chapter 9 - XYZ Custom Plus

9.4 Linear Equations in One Variable
541
B
Equations Involving Fractions
We will now solve some equations that involve fractions. Because integers are
usually easier to work with than fractions, we will begin each problem by clearing
the equation we are trying to solve of all fractions. To do this, we will use the
multiplication property of equality to multiply each side of the equation by the
LCD for all fractions appearing in the equation. Here is an example.
Example 5
Solve the equation } 2
x } 1 }6
x } 5 8.
5. Solve: } 3
x } 1 }6
x } 5 9
x x
Solution The LCD for the fractions } } 2 and }6 } is 6. It has the property that both
2 and 6 divide it evenly. Therefore, if we multiply both sides of the equation by 6,
we will be left with an equation that does not involve fractions.
6 1
} 2
x } 1 }6
x }2 5 6(8) Multiply each side by 6
6 1
} 2
x }2 1 6 1
} 6
x }2 5 6(8) Apply the distributive property
3x 1 x 5 48 Multiplication
4x 5 48 Combine similar terms
x 5 12 Divide each side by 4
We could check our solution by substituting 12 for x in the original equation. If
we do so, the result is a true statement. The solution is 12.
As you can see from Example 5, the most important step in solving an equation
that involves fractions is the first step. In that first step we multiply both sides of
the equation by the LCD for all the fractions in the equation. After we have done
so, the equation is clear of fractions because the LCD has the property that all the
denominators divide it evenly.
Example 6
Solve the equation 2x 1 } 1 2 } 5 }3 4 }.
6. Solve: 3x 1 } 1 4 } 5 }5 8 }
Solution
This time the LCD is 4. We begin by multiplying both sides of the
equation by 4 to clear the equation of fractions.
4 1
2x 1 } 1 2 } 2 5 4 1 }3 4 } 2
4(2x) 1 4 1
} 1 2 } 2 5 4 1 }3 4 } 2
8x 1 2 5 3
8x 5 1
Multiply each side by the LCD, 4
Apply the distributive property
Multiplication
Add 22 to each side
x 5 } 1 } Divide each side by 8
8
7. Solve: } 4 x } 1 3 5 }1 1
}
5
Example 7
Solve for x: } 3 x } 1 2 5 }1 }. (Assume x is not 0.)
2
Solution This time the LCD is 2x. Following the steps we used in Examples 5
and 6, we have
2x 1
} 3 x } 1 2 2 5 2x 1 }1 2 } 2
Multiply through by the LCD, 2x
2x 1
} 3 x } 2 1 2x(2) 5 2x 1 }1 2 } 2
Distributive property
6 1 4x 5 x Multiplication
6 5 23x Add 24x to each side
22 5 x Divide each side by 23
Answers
5. 18 6. } 1 } 7. 25
8