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Chapter 8 Rational Expressions

8.1 Simplifying Rational Expressions 8.2 Multiplying and Dividing Rational Expressions 8.3 Finding the Least Common Denominator 8.4Adding and Subtracting Rational Expressions Putting It All Together 8.5Simplifying Complex Fractions 8.6Solving Rational Equations 8.7Applications of Rational Equations 8 Rational Expressions

Solving Rational Equations 8.6 A rational equation is an equation that contains a rational expression. Some examples of rational equations are Differentiate Between Rational Expressions and Rational Equations

Example 1 Determine whether each is an equation or is a sum or difference of expressions.Then, solve the equation or find the sum or difference. Solution This is an equation because it contains an = sign. We will solve for w using the method we learned in Chapter3: eliminate the denominators by multiplying by the LCD of all of the expressions. The LCD of 3,4 and 12 is 12. Multiply by LCD of 12 and eliminate denominators. Distribute and eliminate denominators. Combine like terms The solution set is {-5}. Subtract 48 on both sides of equation. Divide by 7 on both sides.

Example 2 Determine whether each is an equation or is a sum or difference of expressions.Then, solve the equation or find the sum or difference. Solution This is an not an equation to be solved because it does not contain an = sign. It is a sum of rational expressions. Rewrite each expression with the LCD, then subtract, keeping the Denominators while performing operations. The LCD for 3 and 4 is 12. Rewrite each expression with a denominator of 12. Add the numerators. Distribute. Combine like terms.

Solve Rational Equations

Example 3 Solution Eliminate the denominators by multiplying the equation by the LCD of all the expressions. LCD = 18b. 6 Check: b = 12 Multiply both sides of the equation by the LCD, 18b. Distribute and divide out common factors. Subtract 6b. Factor Set each factor equal to zero. Check: b = -6 The solution set is {-6, 12}. It is very important to check the proposed solution. Sometimes, they might not be solutions.

Since x = -7 makes denominator equal to zero, -7 cannot Be a solution to the equation. Therefore, this equation has No solution. The solution set is Example 4 Solution Eliminate denominators by multiplying every term by the LCD. LCD is x +7. Multiply both sides of the equation by the LCD, x + 7. Distribute and divide out common factors. Multiply. Combine like terms Subtract 3x. Divide by -2. Check: Substitute -7 for x in the original equation.

Example 5 Solution Factor the denominator. Multiply both sides of the equation by the LCD, 4(a+2)(a-2). Distribute and divide out common factors. Continued on next slide…

Multiply. Distribute. Combine like terms. Subtract a and subtract 18. Factor. Set each factor equal to zero. Solve. Look at the factored form of the equation. If a =7, no denominator will equal zero. If a = -2, however, two of the denominators will equal zero. Therefore, we must reject a = -2 as a solution. Check only a = 7. Substitute a = 7 into original equation. The solution set is {7}.

Solve a Proportion Example 6 Solution This rational equation is also a proportion. A proportion is a statement that two ratios are equal. We can solve this proportion as we have solved the other equations in this section, by multiplying both sides of the equation by the LCD. Or, recall from section 3.6 that we can solve a proportion by setting the cross products equal to each other. Multiply. Set the cross products equal to each other. Distribute. Subtract 6c and add 24. The proposed solution, c = 3, does not make a denominator equal zero. Check to verify that the solution set is {3}. Divide by 26.

Solve an Equation for a Specific Variable Example 7 Solution Note that the equation contains a lowercase d and an uppercase D. These represent different quantities, so be sure to write them correctly. Put d in a box. Since d is in the denominator of the rational expression, multiply both sides of the equation by d-D to eliminate the denominator. Put t in a box. Multiply both side by t – T to eliminate the denominator. Distribute ADD Tk. Divide by k.

Example 8 Solution Put z in a box. The LCD of all of the fractions is xyz. Multiply both sides of the equation by xyz. Put z in a box. Multiply both side by LCD, xyz to eliminate the denominator. Factor z out. Divide by (y+x) Divide out common factors.