Download solving rational inequalities

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Math Learning Center
SOLVING RATIONAL INEQUALITIES
West Campus, HS1 203
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CUT-POINT METHOD
To solve a non-linear inequality:
1. Set any denominators = 0 and solve for x. These numbers are called Domain Cut Points. (DCP)
2. Replace the inequality with = and solve for x. These numbers are called Solution Cut Points. (SCP)
3. Draw a number line and label the cut-points as follows:
a. Always label Domain Cut Points with an open circle.
b. Solution Cut-Points
i. If the problem contains ≤ or ≥, use closed circles.
ii. If the problem contains < or >, use open circles.
4. Select a number in each interval and test the original inequality
a. If the inequality is true, shade the interval containing the test value.
b. If the inequality is false, cross out the interval containing the test value.
5. Express the shaded intervals using interval notation.
Example:
Solve.
𝑥𝑥−8
≤ 3 − 𝑥𝑥
𝑥𝑥
𝑥𝑥−8
𝑥𝑥
= 3 − 𝑥𝑥
𝑥𝑥
1
Set any denominators equal to zero.
Domain Cut Point: x = 0
(put an open circle at 0 on the number line)
Replace the inequality with an =.
Write the right side as a fraction
Test
(over 1) and then cross multiply, or clear
5
Numbers: -3
-1
1
fractions by multiplying by the LCD, which is x.
(x – 8)(1) = (x)(3 – x)
For Interval (-∞,-2] : Testing -3
(−3)−8
2
x – 8 = 3x – x Put the equation in standard
≤ 3 − (−3)
3.67 ≤ 6
(−3)
2
2
+x
+x quadratic form.
True
𝑥𝑥−8
=
3−𝑥𝑥
x2 + x – 8 = 3x
-3x
-3x
2
x – 2x – 8 = 0 Factor or use the quadratic formula.
(x + 2)(x – 4)= 0 Set each factor equal to zero and solve for x.
x = -2
x = 4 Solution Cut Points: x = -2 and x = 4
Now put these on a number line and
use closed circles.
For Interval [-2, 0) : Testing -1
(−1)−8
(−1)
≤ 3 − (−1)
For Interval (0, 4] : Testing 1
(1)−8
(1)
(5)
False
≤ 3 − (1)
-7 ≤ 3
≤ 3 − (5)
-0. 6 ≤ −2
For Interval [4, ∞) : Testing 5
(5)−8
-9 ≤ 4
Answer: (-∞,-2] U (0, 4]
True
False