Converting Rational Expressions to Linear Expressions
Some equations are almost linear equations; after one or more steps these equations become linear equations. In this section, we will be converting rational expressions (one quantity divided by another quantity) into linear expressions and square root equations into linear equations. The solution(s) to these converted equations might not be the same as the solution(s) to the original equation. After certain operations, you must check the solution(s) to the converted equation in the original equation.
To solve a rational equation, clear the fraction. In this book, two approaches will be used First, if the equation is in the form of “fraction = fraction,” cross multiply to eliminate the fraction. Second, if there is more than one fraction on one side of the equal sign, the LCD will be determined and each side of the equation will be multiplied by the LCD. These are not the only methods for solving rational equations.
The following is a rational equation in the form of one fraction equals another. We will use the fact that for b and d nonzero, if and only if ad = bc .
This method is called cross multiplication .
Check: is a true statement, so x = 3 is the solution.
Extraneous Solutions
Anytime you multiply (or divide) both sides of the equation by an expression with a variable in it, you must check your solution(s) in the original equation. When you cross multiply, you are implicitly multiplying both sides of the equations by the denominators of each fraction, so you must check your solution in this case as well. The reason is that sometimes a solution to the converted equation will cause a zero to be in a denominator of the original equation. Such solutions are called extraneous solutions . See what happens in the next example.
But x = –2 leads to a zero in a denominator of the original equation, so x = –2 is not a solution to the original equation. The original equation has no solution.
Have you ever wondered why expressions like are not numbers? Let us see what complications arise when we try to see what “ ” might mean. Say
Now cross multiply.
2(1)= 0( x )
Multiplication by zero always yields zero, so the right hand side is zero.
2 = 0 No value for x can make this equation true.
Or, if you try to “clear the fraction” by multiplying both sides of the equation by a common denominator, you will see that an absurd situation arises here, too.
So, 0 = 0 x , which is true for any x . Actually, the expression is not defined.
Simplifying Square Roots
On some equations, you will want to raise both sides of the equation to a power in order to solve for x . Be careful to raise both sides of the equation to the same power, not simply the side with the root. Raising both sides of an equation to an even power is another operation which can introduce extraneous solutions. To see how this can happen, let us look at the equation x = 4. If we square both sides of the equation, we get the equation x ^{2} = 16. This equation has two solutions: x = 4 and x = –4.

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