Linear Equations and Inequalitie
Solving Inequalities
Absolute Value Inequalities
Graphing Equivalent Fractions Lesson Plan
Investigating Liner Equations Using Graphing Calculator
Graphically solving a System of two Linear Equatio
Shifting Reflecting Sketching Graph
Graphs of Rational Functions
Systems of Equations and Inequalities
Graphing Systems of Linear Equat
Solving Inequalities with Absolute Values
Solving Inequalities
Solving Equations & Inequalities
Graph the rational function
Inequalities and Applications
Using MATLAB to Solve Linear Inequalities
Equations and Inequalities
Graph Linear Inequalities in Two Variables
Solving Equations & Inequalities
Teaching Inequalities:A Hypothetical Classroom Case
Graphing Linear Inequalities and Systems of Inequalities
Inequalities and Applications
Solving Inequalities
Quadratic Inequalities
Solving Systems of Linear Equations by Graphing
Systems of Equations and Inequalities
Graphing Linear Inequalities
Solving Inequalities
Solving Inequalities
Solving Equations Algebraically and Graphically
Graphing Linear Equations
Solving Linear Equations and Inequalities Practice Problems
Graphing Linear Inequalities
Equations and Inequalities
Solving Inequalities

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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

■ 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

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.

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 _____________________.

Solve |3x + 7| ≤ 8. Then graph.

Solve |5x – 2| > 3. Then graph.

Inequalities in Two

■ Graphs of Linear Inequalities

■ Systems of Linear Inequalities

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

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.

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.

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.

Graph -1 < y < 5

The graph of the system


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
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
x + y = 3,
5x – y = –27
An (x,y) pair;
one point
System of
x – y ≥ 2;
x ≤3; y ≥ – x
A set of (x,y) pairs;
a region in the plane