# 7.2 – Rational Expressions

## Notes

• A rational number is a quotient, or ratio, of two integers.
• Rational number examples
• Ex 1. $12=\frac{12}{1}$
• Ex 2. $-5=\frac{-5}{1}$
• Ex 3. $\sqrt{\frac{16}{9}}=\frac{4}{3}$
• A rational expression contains numbers or variables separated by a plus or minus sign.
• Rational expression ex
• Ex 1. $\frac{12-4}{3-12}$
• Ex 2. $\frac{12-x}{3x-6}$

## Notes

Rational Expressions
• Equal to Zero
• A rational expression is equal to zero when its numerator is zero.
• Undefined
• A rational expression is undefined whenever its denominator is zero.

## Examples

 Ex 1. For what value of x is a rational expression undefined? Answer: When the denominator is equal to 0 Ex 2. For what value of x is a rational expression below equal to zero? Answer: When the numerator is equal to 0 Ex 3. What is the value of the rational expression below when x is equal to 3? $\frac{12-x}{x-6}$ Substitute 3 for x: $\frac{12-3}{3-6}$ Simplify: $\frac{9}{-3}$ Answer: -3 Ex 4. For what value of x is the rational expression below equal to zero? $\frac{x-4}{x-6}$ Setup: x – 4 = 0 Do the inverse operation (add 4 to both sides) Answer: x = 4 Ex 5. For what value of x is the rational expression below undefined? $\frac{x-4}{4+x}$ Setup: 4 + x = 0 Do the inverse operation (subtract 4 from both sides) Answer: x = -4 Ex 6. For what values of x is the rational expression below undefined? $\frac{x-7}{x^{2}-7x-8}$ Setup: Set the denominator equal to zero and solve. Try factoring (FOIL): $x^{2}-7x-8=0$ $(x-8)(x+1)$ works! Set the first factor equal to zero: $x-8=0$ Add 8 to both sides: x = 8 Set the second factor equal to zero: $x+1=0$ Subtract 1 from both sides.  Answer: x = -1 Ex 7. For what values of x is the rational expression below undefined? $\frac{x+5}{3x^{2}-3}$ Setup: Set the denominator equal to zero and solve. Factor out a 3: $3(x^{2}-1)=0$ Add 1 to both sides: $x^{2}=1$ Set the variable factor equal to zero: $x^{2}=1$ Take the square root of both sides: $\sqrt{x^{2}}=\sqrt{1}$ Answer: $x\pm 1$