aim: rational inequalities course: adv. alg. & trig. aim: how do we solve various inequality...
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Aim: Rational Inequalities Course: Adv. Alg. & Trig.
Aim: How do we solve various inequality problems?
Do Now:
Graph the solution set of
3x + 5 > 14
Aim: Rational Inequalities Course: Adv. Alg. & Trig.
Rational Inequalities
34
x
x
34
x
x
What values of x makes this statement true?
3 4x x
12 3x x
4 12x
3x
cross-multiply?
distribute
isolate variable
divide
check:2
3 1 3 true4 2
53 5 3
4 5
x = 5
also true!
Aim: Rational Inequalities Course: Adv. Alg. & Trig.
Rational Inequalities
34
x
x
34
x
x
What values of x makes this statement true?
3 04
x
x
standard form to
find zeros
3 40
4 4
xx
x x
subtract -
LCD
3 40
4
x x
x
Combine & Simplify
12 3 4 120
4 4
x x x
x x
Find zeros of Numerator & Denominator
4x – 12 = 0
4 – x = 0
x = 3
x = 4
critical values
Aim: Rational Inequalities Course: Adv. Alg. & Trig.
Rational Inequalities - Algebraically
Test Interval
Representative x-value
Value of Polynomial
(-, 3) x = 0 y = -3
(3, 4) x = 3.5(4, ) x = 10
y = 4
y = -4 2/3
Find zeros of Numerator & Denominator
4x – 12 = 0
4 – x = 0
x = 3
x = 4
critical values
34
x
x
3 0
4
x
x
4 12
04
x
x
-10 -5 5 100
34
x
x
( ,3) (4, )
()
Aim: Rational Inequalities Course: Adv. Alg. & Trig.
Rational Inequalities - Algebraically
2x 7
x 53
Algebraically, find the zeros, the critical numbers and test intervals of
2x 7
x 5 3 0 Standard form
to find zeroes
2x 7 3x 15
x 50 Add fractions
x 8
x 50 Simplify
to find zeroes
critical numbers: x = 8; x = 5
Note: rational expressions change sign at its zeros and its undefined values.
=
Aim: Rational Inequalities Course: Adv. Alg. & Trig.
Rational Inequalities - Algebraically
2x 7
x 53
Algebraically find the zeros, the critical numbers and test intervals of
x 8
x 50 critical numbers: x = 8
x = 5
Test Interval
Representative x-value
Value of Polynomial
(-, 5) x = 0 y = -8/5
(5, 8] x = 6[8, ) x = 10
y = 2
y = -2/5
2x 7
x 53
( ,5) [8, )
Aim: Rational Inequalities Course: Adv. Alg. & Trig.
2
-2
-4
5 10
h x = 2x-7
x-5-3
Rational Inequalities
2x 7
x 53Solve
2x 7
x 5 3 0
Standard form
Graphically:
y is negative from(-, 5) and [8, )
2x 7
x 5 3 y
(8, 0)
< 0
Aim: Rational Inequalities Course: Adv. Alg. & Trig.
Using the Graphing Calculator
2x 7
x 53
x 8
x 50
x 8
x 5Y1
= (-x + 8)/(x – 5) Y1
2nd MATH 6 GRAPH0
≤ 0
Aim: Rational Inequalities Course: Adv. Alg. & Trig.
Model Problem
1 31
2 4t t Solve
Test Interval
Representative t-value
Value of Polynomial
(-, 0)(0, 1.25]
[1.25, )
t = -1t = 1
t = 10
f(t) = -9/4f(t) = 1/4
f(t) = -36/40
Note: rational expressions change sign
at its zeros and its undefined values.
1 31 0
2 4t t
1 34 1 0
2 4t
t t
5 4 0
1 25.
t
t
==
1 31 0
2 4t t
critical numbers: t = 0; t = 1.25
2
-2
(1.25, 0)(0, 0)