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 Dependent Variable

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# Inequalities and Applications

Interval Notation and Domains

Write the domain of f (x) in set notation,
interval notation and graph the set. Absolute-Value Equations
and Inequalities

■ Equations with Absolute Value
4.3

■ Inequalities with Absolute Value

Absolute Value

 The absolute value of x, denoted |x|, is defined as: (When x is nonnegative, the absolute value of x is x. When x is negative, the absolute value of x is the opposite of x.)

Example Find the solution set:
a) |x| = 6; b) |x| = 0; c) |x| = –2

Absolute-Value Principle for Equations

 For any expression X and any positive number p: a) The solutions of |X| = p must . b) The equation |X| = 0 is equivalent to the equation c) The equation |X| = –p has . d) The equation |X| = |Y| means .

Example Find the solution set:
a) |2x +1| = 5; b) |3 – 4x| = –10

Example
Given f (x) = 3|x+5| – 4, find all x for which f (x) = 11.

Absolute Value Equations | X | = | Y |
Sometimes an equation has two absolute-values, like| X | = | Y |.
This means that X and Y are the same distance from zero.

If X and Y are the same distance from zero,
then either they’re the same number or they are opposites.

Example
To solve |3x – 5| = |8 + 4x| consider the two cases:

Inequalities with Absolute Value
Our methods for solving equations with absolute value
can be adapted for solving inequalities.
Solve and graph.
|x| < 3 |x| > 3

Absolute Value Equations & Inequalities

 Let Stuff be any algebraic expression, then the solutions of: a) |Stuff | = p are x’s where _____________________. b) |Stuff | < p are x’s where _____________________. c) |Stuff | > p are x’s where _____________________. Example
Solve |3x + 7| ≤ 8. Then graph.

Example
Solve |5x – 2| > 3. Then graph.

Inequalities in Two
Variables

■ Graphs of Linear Inequalities
4.4

■ Systems of Linear Inequalities

Graphs of Linear Inequalities
A linear inequality is formed with <, ≤, >, ≥.
Solutions of linear inequalities are ordered pairs.

Example
Determine whether (1, 5) and (6, –2) are
solutions of the inequality 3x – y < 5.

Graphing Linear Inequalities
The graph of a linear equation is a straight line.
The graph of a linear inequality is a half-plane,
with a boundary that is a straight line. Graphing Linear Inequalities

1) ___________________________________
A. Decide if the line is included in the solutions set or not.
■ < or > means the line is not in solutions – dashed line
■ ≤ or ≥ means the line is a solutions – solid line
B. Make the inequality into an equation (=) and graph it.

2)____________________________________
1. Pick a test point not on the line
2. Plug it into the original inequality and evaluate true or false
3. Shade the side containing the point that make inequality true.

Example
Graph 2x + 5y > 15 Systems of Linear Inequalities
Graph the system x − y ≤ 3 and x + y > 3. To graph a system of inequalities:
1) graph each inequality and
2) find the intersection of the individual graphs.

Example
Graph -1 < y < 5 -3

Example
The graph of the system  Example  REVIEW – Six types of problems

 Type Example Solution Graph Linear Equation in one variable 2x – 8 = 3(x + 5) One number Linear Inequalities in one variable –3x + 5 > 2 A set of numbers; an interval Linear Equation in two variables 2x + y = 5 A set of (x,y) pairs; a line Linear Inequalities in two variables x + y ≥ 4 A set of (x,y) pairs; a half-plane System of Equations x + y = 3, 5x – y = –27 An (x,y) pair; one point System of Inequalities x – y ≥ 2; x ≤3; y ≥ – x A set of (x,y) pairs; a region in the plane