Linear Inequalities

Lesson 2.4

2

Inequalities

Definition Start with an equation 3x + 5 = 17 Replace the equals sign with one of

Examples

2 5 7x

8 2 3 6x x

Note that each of these can be changed into the form ax + b § 0

where the § is the required inequality symbol

Note that each of these can be changed into the form ax + b § 0

where the § is the required inequality symbol

3

Solving Inequality in One Variable – Symbolic Method

Treat inequality in similar manner as an equation Add same thing to both sides of inequality Multiply both sides by same positive value If negative value used, inequality reverses

Try it!

Inequality remains same

<

3 6 9x

4

Intersection of Graphs Method

Given f(x) > g(x) Graph both f(x) and g(x) Find the point of intersection

The solution set includes thex-values where f(x) is above g(x)

•

f(x)g(x)

5

Try It Out

Given

Graph each side of the equation Find intersection Decide which side of the intersection matches

2 10.5 13.7x x

6

X-Intercept Method

Write the inequality as h(x) < 0

Graph y = h(x)

Solutions occur where graph is below x-axis

7

Numerical Method

Write the inequality as h(x) < 0

Place h(x) in Y= screen

Use Tables (♦Y) and observe values of x where the function is less than 0

Assignment

Lesson 2.4A Page 142 Exercises 1 – 73 EOO

8

9

Compound Inequality

Definition:Two inequalities connected by the words and or or

Examples5 12x or x

0 7x and x

3 1 2x x

Anything in this format should be considered an "and" compound inequality

Anything in this format should be considered an "and" compound inequality

10

Interval Notation

Solution displayed in interval notation [-1, 2)

The [ or ] means that the number is included in the interval

The ( or ) means that the endpoint is not part of the interval

[ )

11

Solving Compound Inequality

Symbolically Add the same thing to each part Multiply each part by same positive number

Justify each step in thesequence to the right

3 3 6 12

3 3 6

1 2

x

x

x

12

Solving Compound Inequality

Graphically Graph all three parts

of the inequalities asfunctions

The solution willbe the values ofx for which the middle function isbetween the other two

2 50.2 8

3

xx

Try It Out

Temp in Fahrenheit x miles above ground level given by T(x) = 85 – 19x

Use intersection of graphs todetermine when temp is < 32F

What does x-intercept on graphy = T(x) represent?

13

Linear Regression

Home ownership rates given by table

Determine a linear modeling function Estimate years when ownership percent was

between 58% and 60% Do we interpolate or extrapolate?

14

x 1900 1950 1980 2006

P 47% 55% 64% 69%

15

Assignments

Lesson 2.4B Page 144 Exercises 83 – 99 EOO