Upload
pemey13
View
102
Download
3
Embed Size (px)
Citation preview
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Solve rational equations and inequalities.
Objective
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
rational equationextraneous solutionrational inequality
Vocabulary
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
A rational equation is an equation that contains one or more rational expressions. The time t in hours that it takes to travel d miles can be determined by using the equation t = , where r is the average rate of speed. This equation is a rational equation.
dr
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
To solve a rational equation, start by multiplying each term of the equation by the least common denominator (LCD) of all of the expressions in the equation. This step eliminates the denominators of the rational expression and results in an equation you can solve by using algebra.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Solve the equation x – = 3.
Example 1: Solving Rational Equations
18x
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
x – = 3 18xCheck
1866 – 3
33
6 – 33
x – = 3
18(–3)(–3) – 3
33
–3 + 63
18x
Example 1 Continued
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 1a
Solve the equation = + 2. 4x
103
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 1b
Solve the equation + = – . 54
6x
74
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 1c
Solve the equation x = – 1. 6x
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
An extraneous solution is a solution of an equation derived from an original equation that is not a solution of the original equation. When you solve a rational equation, it is possible to get extraneous solutions. These values should be eliminated from the solution set. Always check your solutions by substituting them into the original equation.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Solve each equation. Example 2A: Extraneous Solutions
5x x – 2
3x + 4 x – 2 =
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check Substitute 2 for x in the original equation.
5x x – 2
3x + 4 x – 2 =
5(2) 2 – 2
3(2) + 4 2 – 2
100
100
Division by 0 is undefined.
Example 2A Continued
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Solve each equation. Example 2B: Extraneous Solutions
2x – 5 x – 8
11 x – 8 + =x
2
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
2x – 5 x – 8
11 x – 8 + =x
2 2x – 5
x – 8 11 x – 8 + – = 0. x
2
CheckWrite
as
Graph the left side of the equation as Y1. Identify the values of x for which Y1 = 0.The graph intersects the x-axis only when x = –4. Therefore, x = –4 is the only solution.
Example 2B Continued
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 2a Solve the equation . 16
x2 – 16 2
x – 4 =
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
x x – 1 Solve the equation . 1
x – 1 = + x6
Check It Out! Example 2b
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
A rational inequality is an inequality that contains one or more rational expressions. One way to solve rational inequalities is by using graphs and tables.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Example 5: Using Graphs and Tables to Solve Rational Equations and Inequalities
Solve ≤ 3 by using a graph and a table. xx – 6
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Example 5: Using Graphs and Tables to Solve Rational Equations and Inequalities
Solve ≤ 3 by using a graph and a table. xx – 6
xx – 6
Use a graph. On a graphing calculator, Y1 = and Y2 = 3.
The graph of Y1 is at or below the graph of Y2 when x < 6 or when x ≥ 9.
(9, 3)
Vertical asymptote:
x = 6
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Example 5 Continued
Use a table. The table shows that Y1 is undefined when x = 6 and that Y1 ≤ Y2 when x ≥ 9.
The solution of the inequality is x < 6 or x ≥ 9.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 5a Solve ≤ 4 by using a graph and a table. x
x – 3
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 5a Solve ≤ 4 by using a graph and a table. x
x – 3
xx – 3
Use a graph. On a graphing calculator, Y1 = and Y2 = 4.
The graph of Y1 is at or below the graph of Y2 when x < 3 or when x ≥ 4.
(4, 4)
Vertical asymptote:
x = 3
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 5a continued
Use a table. The table shows that Y1 is undefined when x = 3 and that Y1 ≤ Y2 when x ≥ 4.
The solution of the inequality is x < 3 or x ≥ 4.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 5b
Solve = –2 by using a graph and a table. 8x + 1
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 5b
Solve = –2 by using a graph and a table. 8x + 1
The graph of Y1 is at or below the graph of Y2 when x = –5.
(–5, –2)
Vertical asymptote:
x = –1
8x + 1
Use a graph. On a graphing calculator, Y1 = and Y2 = –2.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Use a table. The table shows that Y1 is undefined when x = –1 and that Y1 ≤ Y2 when x = –5.
The solution of the inequality is x = –5.
Check It Out! Example 5b continued
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
You can also solve rational inequalities algebraically. You start by multiplying each term by the least common denominator (LCD) of all the expressions in the inequality. However, you must consider two cases: the LCD is positive or the LCD is negative.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Example 6: Solving Rational Inequalities Algebraically
Solve ≤ 3 algebraically. 6x – 8
Case 1 LCD is positive.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Example 6: Solving Rational Inequalities Algebraically
Solve ≤ 3 algebraically. 6x – 8
Case 2 LCD is negative.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 6a
Solve ≥ –4 algebraically. 6x – 2
Case 1 LCD is positive.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 6a Continued
Solve ≥ –4 algebraically. 6x – 2
Case 2 LCD is negative.
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Case 1 LCD is positive.
Solve < 6 algebraically. 9x + 3
Check It Out! Example 6b
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Check It Out! Example 6b Continued
Case 2 LCD is negative.
Solve < 6 algebraically. 9x + 3
Holt McDougal Algebra 2
Solving Rational Equationsand Inequalities
Lesson Quiz
2.
1. x + 2 x
x – 1 2 =
6xx + 4
7x + 4 x + 4 =
3.
4.
x + 2 x – 3
5 x – 3 + =x
5
Solve each equation or inequality.
4x – 3
5. A college basketball player has made 58 out of 82 attempted free-throws this season. How many additional free-throws must she make in a row to raise her free-throw percentage to 90%?
≥ 2