Week 15 – Precalc 11

One of the things we learned this week in Precalc 11 is how to solve rational equations.

Cross-Multiplying

The cross-multiplying method only works if a fraction is equal to another fraction (\frac{2x-1}{4}=\frac{4x+2}{2}) and does not work with 3 or more terms (\frac{2x-1}{4} + \frac{3-2}{x}=\frac{4x+2}{2})

Examples

Solve: \frac{x+3}{2x}=\frac{x-5}{3}
First we need to determine the non-permissible values for x.
x\neq 0
Next, cross-multiply. Multiply the numerator for the first fraction to the denominator for the second one, vice versa with the denominator for the first fraction.
3(x+3)=2x(x-5)
Then distribute.
3x+9=2x^2-10x
Since there is a square, we know that it’s a quadratic and it has 2 possible solutions. Move all the terms on one side.
0=2x^2-13x-9
Since we it doesn’t factor with nice numbers we can use the quadratic formula.
x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}
x=\frac{-(-13)\pm \sqrt{(-13)^2-4(2)(-9)}}{2(2)}
x=\frac{13\pm \sqrt{241}}{4}
x_1=\frac{13+\sqrt{241}}{4}
x_2=\frac{13-\sqrt{241}}{4}

Solve: \frac{2x-3}{x+1}={x+6}{x-2}
x\neq -1, 2
(x-2)(2x-3)=(x+6)(x+1)
2x^2-3x-4x+6=x^2+x+6x+6
2x^2-7x+6=x^2+7x+6
x^2-14x=0
x(x-14)=0
x_1=0
x_2=14

Converting to a common denominator

Works everytime even with 3 or more terms.

Example

Solve: 3x- \frac{1}{x}=\frac{1}{2}
First determine the non-permissible values.
x\neq 0
Then determine the Lowest/Least Common Denominator (LCD). LCD=2x
Then multiply the fractions so that they have the same denominators.
\frac{3x(2x)}{1(2x)} - \frac{1(2)}{x(2)}=\frac{1(x)}{2(x)}
\frac{6x^2}{2x} - \frac{2}{2x}=\frac{x}{2x}
Since the denominators are all the same the numerators also have to be the same, so we could ignore the denominators.
6x^2-2=x
Since it’s a quadratic we have to make it equal 0 and then factor.
6x^2-x-2=0
(2x+1)(3x-2)=0
x_1=- \frac{1}{2}
x_2=\frac{2}{3}

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