Transcript rational equation

5-5
Solving Rational Equations
and Inequalities
Warm Up
Find the least common multiple for each
pair.
1. 2x2 and 4x2 – 2x
2. x + 5 and x2 – x – 30
Add or subtract. Identify any x-values for
which the expression is undefined.
1
1
3.
+
x–2
4x
4.
1
1
–
x
x2
Holt McDougal Algebra 2
5-5
Solving Rational Equations
and Inequalities
Objective
Solve rational equations and
inequalities.
Vocabulary
rational equation
extraneous solution
rational inequality
Holt McDougal Algebra 2
5-5
Solving Rational Equations
and 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 = d , where r is the average rate
r
of speed. This equation is a rational equation.
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
5-5
Solving Rational Equations
and Inequalities
Example 1: Solving Rational Equations
Solve the equation x – 18 = 3.
x
Holt McDougal Algebra 2
5-5
Solving Rational Equations
and 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
5-5
Solving Rational Equations
and Inequalities
Example 2A: Extraneous Solutions
Solve each equation.
5x = 3x + 4
x–2
x–2
Holt McDougal Algebra 2
2x – 5 + x =
11
x–8
x–8
2
5-5
Solving Rational Equations
and Inequalities
Example 3: Problem-Solving Application
A jet travels 3950 mi from Chicago, Illinois, to
London, England, and 3950 mi on the return trip. The
total flying time is 16.5 h. The return trip takes
longer due to winds that generally blow from west to
east. If the jet’s average speed with no wind is 485
mi/h, what is the average speed of the wind during
the round-trip flight? Round to the nearest mile per
hour.
Holt McDougal Algebra 2
5-5
Solving Rational Equations
and Inequalities
Example 4: Work Application
Natalie can finish a 500-piece puzzle in about 8 hours.
When Natalie and Renzo work together, they can finish
a 500-piece puzzle in about 4.5 hours. About how long
will it take Renzo to finish a 500-piece puzzle if he
works by himself?
Holt McDougal Algebra 2
5-5
Solving Rational Equations
and 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
5-5
Solving Rational Equations
and Inequalities
Example 5: Using Graphs and Tables to Solve
Rational Equations and Inequalities
Solve
x
x– 6
≤ 3 by using a graph and a table.
(9, 3)
Vertical
asymptote:
x=6
Holt McDougal Algebra 2
5-5
Solving Rational Equations
and 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
5-5
Solving Rational Equations
and Inequalities
Example 6: Solving Rational Inequalities
Algebraically
Solve
6
x– 8
≤ 3 algebraically.
Pg 353 20-28e, 29,30,32-37, 38-46e, 47-49, 54-56, 58-61
Holt McDougal Algebra 2