Download solving rational inequalities

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SOLVING RATIONAL INEQUALITIES
TRADITIONAL METHOD
To solve a non-linear inequality:
1. If needed, rewrite the inequality so that 0 is on one side. This usually requires you to add or subtract
term(s) to both sides of the inequality.
2. If needed, simplify the non-zero side so that there is a single fraction. This usually requires finding a
common denominator.
3. Set the numerator equal to zero and solve. These values determine the intervals on the number line.
Use:
a. closed circles if the problem contains ≥ or ≤.
b. open circles if the problem contains > or <.
4. Set the denominator equal to zero and solve. These values determine the intervals on the number
line. Always use open circles when you put these values on the number line.
5. 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.
6. Express the shaded intervals using interval notation.
Example:
Solve.
𝑥𝑥−8
≤ 3 − 𝑥𝑥
𝑥𝑥
𝑥𝑥−8
𝑥𝑥
𝑥𝑥−8
𝑥𝑥
Subtract 3 and add x to both sides to
get the right side to be zero.
− 3 + 𝑥𝑥 ≤ 0
−
3𝑥𝑥
𝑥𝑥
+
𝑥𝑥−8−3𝑥𝑥+𝑥𝑥 2
𝑥𝑥
𝑥𝑥 2−2𝑥𝑥−8
𝑥𝑥
x=0
𝑥𝑥 2
𝑥𝑥
Build fractions with a common
denominator.
≤ 0 Write the left side as a single fraction
over the common denominator.
≤0
≤0
x2 – 2x – 8 = 0
Test
Numbers: -3
Then combine like terms in the
numerator.
Set the denominator equal to zero
and solve.
Put this value on the number line
as an open circle.
Set the numerator equal to zero and solve.
Factor or use the quadratic formula.
(x + 2)(x – 4)= 0
Set each factor equal to zero and solve for x.
x = -2
Now put these values on a number
line and use closed circles.
x=4
-1
5
1
For Interval (-∞,-2] : Testing -3
(−3)−8
≤ 3 − (−3)
3.67 ≤ 6
(−3)
For Interval [-2, 0) : Testing -1
(−1)−8
(−1)
≤ 3 − (−1)
For Interval (0, 4] : Testing 1
(1)−8
(1)
(5)
-9 ≤ 4
False
≤ 3 − (1)
-7 ≤ 3
≤ 3 − (5)
-0. 6 ≤ −2
For Interval [4, ∞) : Testing 5
(5)−8
True
Answer: (-∞,-2] U (0, 4]
True
False