Chapter 4 Inequalities Inequality: x < 3 Graph: Interval Notation: Set Builder Notation:

Inequality: x < 3 Graph: Interval Notation: Set Builder Notation:

Inequality: x > 3 Graph: Interval Notation: Set Builder Notation:

Inequality: x > 3 Graph: Interval Notation: Set Builder Notation:

Inequality: -2 < x < 1 This is called a ________________ interval. Graph: Interval Notation: Set Builder Notation:

Inequality: -4 < x < -1 This is called a ________________ interval. Graph: Interval Notation: Set Builder Notation:

a > b

a + c > b + c (all c)

a – c > b - c (all c)

ac > bc (c >0)

ac < bc (c <0)

ac = bc (c =0)

Solve and graph: 1) 16 – 7y > 10y – 4

2) 3−𝑥

−5≤ −2

3) 6[3r-4(r-1)] > 4r

4) Let f(x) = -3(x+8) - 5x and g(x) = 4x – 9 Find all x for which f(x) > g(x) Note: Mathematical statements are very similar to English statements. Ex: Your teacher is an eggplant. True or False? Ex: True or False: a) 3x < 3x + 1 b) 3x > 3x + 1 What does your answer to the second example mean in the context of the equation as a question?

Solving Compound Inequalities

Intersection: {1, 2, 3} {1, 3, 5} = { }

Union: {1, 2, 3} {1, 3, 5} = { }

Venn Diagram: Intersection “and” member of both sets

Union “or” member of either set (or both)

Ex: Solve and graph 1) -2 < x and x < 1 -2 < x x < 1 -2 < x and x < 1

2) -1 < 2x + 5 < 13 Note: Could you solve -1 < 2x + 5 > 13 ? 3) 2x – 5 > -3 and 5x + 2 > 17

4) 2x – 3 > 1 and -3x + 1 > -2 5) 7 + 2x < -1 or 13 – 5x < 3

6) 3x – 11 < 4 or 4x + 9 > 1

7) - 10 < 𝑥+6

−3< -8

Absolute Value Equations and Inequalities

|3| = 3 When you put in a positive number, you get the same number out |-3| = 3 When you put in a negative number, you get the opposite |0| = 0 When you put in zero, you get zero out |x| = ? We want it to be the same (x) for positive numbers and zero, but the opposite (-x) for negative numbers. How do you write that?

|𝑥| = { 𝑥 𝑖𝑓 𝑥 ≥ 0−𝑥 𝑖𝑓 𝑥 < 0

Ex: Solve

1) |x| = 3

2) | 2x + 5 | = 13

3) 3| x - 2 | = 3

4) 2| x + 4 | = -3

5) 4| x – 3 | + 2 = 3

6) | 2x - 3 | = | x + 5 |

Inequalities with Absolute Value Ex: Solve | x | < 4 First – let’s try plugging in values for x:

x Is it a solution?

0 Yes

1

2

3

4

5

x Is it a solution?

-1

-2

-3

-4

-5

-6

Next – let’s graph the solutions we got on a number line: Ex: Solve | x | > 4 Ex: Solve, Graph and Check:

1) | 3x – 2 | < 4

2) | 4x + 2 | > 6