Exercise 32 Page 695

We are asked to write a rational equation that has 5 as a solution. Let's start by recalling a few important definitions.

• Rational Equation: an equation that contains at least one rational expression.
• Rational Expression: a fraction whose numerator and denominator are polynomials.
• Extraneous Solution: a solution that is derived from the original equation, but it is not actually a solution.

With these definitions in mind, we can write the equation! First, let's arbitrarily choose the variable that we will use to write the equation. We will let x be our variable. We will create our equation by starting from the solution that we want it to have. x= 5In order to obtain a rational equation, we have to use the Properties of Equality to perform the same operations on both sides of the equation. Let's start by adding x and subtracting 6 on both sides.

x=5

AddEqn

LHS+x=RHS+x

2x=x+5

SubEqn

LHS-6=RHS-6

2x-6=x-1

Remember that we must have at least one rational expression in our equation. Therefore, we will divide both sides of the equation by a polynomial. Let x^2-1 be our polynomial. Note that we have to divide the equation by a polynomial that is not equal to 0 when x= 5. Otherwise, x= 5 will be an extraneous solution.

2x-6=x-1

DivEqn

.LHS /(x^2-1).=.RHS /(x^2-1).

2x-6/x^2-1=x-1/x^2-1

We can further simplify the right-hand side of the equation by factoring the denominator and canceling any common factors.

2x-6/x^2-1=x-1/x^2-1

▼

Simplify right-hand side

FacDiffSquares

a^2-b^2=(a+b)(a-b)

2x-6/x^2-1=x-1/(x+1)(x-1)

CancelCommonFac

Cancel out common factors

2x-6/x^2-1=x-1/(x+1)(x-1)

SimpQuot

Simplify quotient

2x-6/x^2-1=1/x+1

We have obtained a rational equation that has 5 as a solution. 2x-6/x^2-1=1/x+1 To be sure, let's check that x= 5 is actually a solution of the equation. This can be done by substituting 5 for x in the equation and simplifying.

2x-6/x^2-1=1/x+1

Substitute

x= 5

2( 5)-6/5^2-1? =1/5+1

▼

Evaluate

CalcPow

Calculate power

2(5)-6/25-1? =1/5+1

Multiply

Multiply

10-6/25-1? =1/5+1

AddSubTerms

Add and subtract terms

4/24? =1/6

ReduceFrac

a/b=.a /4./.b /4.

1/6=1/6 ✓

Since the obtained statement is true, x= 5 is a solution of our rational equation. Please note that there are infinitely many rational equations that have 5 as a solution, so our answer is only an example. Here are some other examples.

Examples
x-5/3x-4=x^2-4x-5
2x-10/x^2+1=x^2-25/x^3-x^2+3
x+1/2x^2-6=x^2-22/x^2-2x+7