M2: Absolute Value Functions, Equations, and Inequalities

Question 1

What is an Absolute Value

Answer 1

Absolute value, written as ⎜x⎟, represents the distance between x and 0 on a number line. As a distance, absolute
value is always non-negative. For every point on a number line (except at 0), there is another point on the opposite
side of 0 that is the same distance from 0. For example, both 5 and –5 are five units away from 0. Thus, ⎜−5⎟ = 5 and
⎜5⎟ = 5.

Question 2

What is the formula for an absolute function?

Answer 2

g(x) = a| 1/b (x-h) | + k

Question 3

How do you find the vertex of an equation?

Problem: y=2(x-7)+3

Answer 3

The vertex is found by taking negative h and positive k
and putting them into point form (-h, k).
h k
| |
V V
y = 2 (x - 7) + 3

The vertex is (7, 3).

Question 4

If the h is negative in the equation, what is it really?

Answer 4

The h is actually positive, it is opposite of what it shows.

Question 5

if the h is positive in the equation, what is it really?

Answer 5

The h is actually negative, it is opposite of what it shows.

Question 6

when the k is positive in the equation, what is it really?

Answer 6

The k is actually positive it is going up, vertical shift

Question 7

when the k is negative in the equation, what is it really?

Answer 7

The k is actually negative it is going down, vertical shift

Question 8

How do you solve each absolute value equation algebraically? Graph the solutions on a number line.
With an example of ⎜3x⎟ + 2 = 8

Answer 8

⎜3x⎟ + 2 = 8
Subtract 2 from both sides. ⎜3x⎟ = 6

Rewrite as two equations. 3x = 6 or 3x = −6

                        Solve for x. x = 2 or x = −2 

-6  -5  -4  -3  -2  -1   0  1  2  3  4  5  6

Question 9

How to isolate the absolute value expression in each equation to determine if it is no solution? With an example of −5 ⎜x + 1⎟ + 2 = 12

Answer 9

−5 ⎜x + 1⎟ + 2 = 12
Subtract 2 from both sides. − 5 ⎜x + 1⎟ = 10

       Divide both sides by −5 . ⎜x + 1⎟ = −2

Absolute values are never negative. No Solution

Question 10

how to solve the inequality algebraically? With an example of ⎜4 - x⎟ + 15 > 21

Answer 10

⎜4 - x⎟ + 15 > 21

⎜4 - x⎟ > 6

4 - x < -6 or 4 - x > 6

• x < -10 or - x > 2

x > 10 or x < -2

Question 11

what is a compound inequality?

Answer 11

Two or more inequalities joined together by “and” or “or”

Question 12

how does an inequality work?

Answer 12

A statement that compares two quantities using , ≤,≥, or ≠ (a comparison that is not equal)

Question 13

what is a disjunction?

Answer 13

a compound statement formed by joining two or more statements with the word or (math statement that uses “or” to combine two statements (only one has to be true)

Question 14

what is a vertex

Answer 14

When the graph begins to change directions