Rational Equations and Inequalities

Holt McDougal Algebra 2
Solving Rational Equations
and Inequalities
Solve rational equations and
inequalities.
Objective

and Inequalities
rational equation
extraneous solution
rational inequality
Vocabulary

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 = , where r is the average rate
of speed. This equation is a rational equation.
d
r

and 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.

and Inequalities
Solve the equation x – = 3.
Example 1: Solving Rational Equations
18
x

and Inequalities
x – = 3
18
x
Check
18
6
6 – 3
3
3
6 – 3
3 
x – = 3
18
(–3)
(–3) – 3
3
3
–3 + 6
3 
18
x
Example 1 Continued

and Inequalities
Check It Out! Example 1a
Solve the equation = + 2.4
x
10
3

and Inequalities
Check It Out! Example 1b
Solve the equation + = – .5
4
6
x
7
4

and Inequalities
Check It Out! Example 1c
Solve the equation x = – 1.6
x

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.

and Inequalities
Solve each equation.
Example 2A: Extraneous Solutions
5x
x – 2
3x + 4
x – 2
=

and 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
10
0
10
0
Division by 0 is
undefined.
Example 2A Continued

and Inequalities
Solve each equation.
Example 2B: Extraneous Solutions
2x – 5
x – 8
11
x – 8
+ =
x
2

and Inequalities
2x – 5
x – 8
11
x – 8
+ =x
2
2x – 5
x – 8
11
x – 8
+ – = 0.x
2
Check
Write
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

and Inequalities
Solve the equation .16
x2
– 16
2
x – 4
=

and Inequalities
x
x – 1
Solve the equation .1
x – 1
= + x
6

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.

and Inequalities
Example 5: Using Graphs and Tables to Solve
Rational Equations and Inequalities
Solve ≤ 3 by using a graph and a table.x
x – 6

and Inequalities
Example 5: Using Graphs and Tables to Solve
Rational Equations and Inequalities
x – 6
x
x – 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

and 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.

and Inequalities
x – 3

and Inequalities
x – 3
x
x – 3
Use a graph. On a
Y1 = and Y2 = 4.
Y2 when x < 3 or
when x ≥ 4.
(4, 4)
Vertical
asymptote:
x = 3

and Inequalities
Check It Out! Example 5a continued
undefined when x = 3 and that Y1 ≤ Y2
when x ≥ 4.
The solution of the inequality is x < 3 or x ≥ 4.

and Inequalities
Solve = –2 by using a graph and a table.8
x + 1

and Inequalities
Solve = –2 by using a graph and a table.8
x + 1
Y2 when x = –5.
(–5, –2)
Vertical
asymptote:
x = –1
8
x + 1
Use a graph. On a
Y1 = and Y2 = –2.

and Inequalities
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

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.

and Inequalities
Example 6: Solving Rational Inequalities
Algebraically
Solve ≤ 3 algebraically.6
x – 8
Case 1 LCD is positive.

and Inequalities
Example 6: Solving Rational Inequalities
Algebraically
Solve ≤ 3 algebraically.6
x – 8
Case 2 LCD is negative.

and Inequalities
Solve ≥ –4 algebraically.6
x – 2

and Inequalities
Check It Out! Example 6a Continued
Solve ≥ –4 algebraically.6
x – 2

and Inequalities
Solve < 6 algebraically.9
x + 3

and Inequalities
Check It Out! Example 6b Continued
Solve < 6 algebraically.9
x + 3

and Inequalities
Lesson Quiz
2.
1. x + 2
x
x – 1
2
=
6x
x + 4
7x + 4
x + 4
=
3.
4.
x + 2
x – 3
5
x – 3
+ =x
5
Solve each equation or inequality.
4
x – 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

Rational Equations and Inequalities

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