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Finding the Greatest Common Factor
Factoring Trinomials
Absolute Value Function
A Summary of Factoring Polynomials
Solving Equations with One Radical Term
Adding Fractions
Subtracting Fractions
The FOIL Method
Graphing Compound Inequalities
Solving Absolute Value Inequalities
Adding and Subtracting Polynomials
Using Slope
Solving Quadratic Equations
Multiplication Properties of Exponents
Completing the Square
Solving Systems of Equations by using the Substitution Method
Combining Like Radical Terms
Elimination Using Multiplication
Solving Equations
Pythagoras' Theorem 1
Finding the Least Common Multiples
Multiplying and Dividing in Scientific Notation
Adding and Subtracting Fractions
Solving Quadratic Equations
Adding and Subtracting Fractions
Multiplication by 111
Adding Fractions
Multiplying and Dividing Rational Numbers
Multiplication by 50
Solving Linear Inequalities in One Variable
Simplifying Cube Roots That Contain Integers
Graphing Compound Inequalities
Simple Trinomials as Products of Binomials
Writing Linear Equations in Slope-Intercept Form
Solving Linear Equations
Lines and Equations
The Intercepts of a Parabola
Absolute Value Function
Solving Equations
Solving Compound Linear Inequalities
Complex Numbers
Factoring the Difference of Two Squares
Multiplying and Dividing Rational Expressions
Adding and Subtracting Radicals
Multiplying and Dividing Signed Numbers
Solving Systems of Equations
Factoring Out the Opposite of the GCF
Multiplying Special Polynomials
Properties of Exponents
Scientific Notation
Multiplying Rational Expressions
Adding and Subtracting Rational Expressions With Unlike Denominators
Multiplication by 25
Decimals to Fractions
Solving Quadratic Equations by Completing the Square
Quotient Rule for Exponents
Simplifying Square Roots
Multiplying and Dividing Rational Expressions
Independent, Inconsistent, and Dependent Systems of Equations
Graphing Lines in the Coordinate Plane
Graphing Functions
Powers of Ten
Zero Power Property of Exponents
The Vertex of a Parabola
Rationalizing the Denominator
Test for Factorability for Quadratic Trinomials
Trinomial Squares
Solving Two-Step Equations
Solving Linear Equations Containing Fractions
Multiplying by 125
Exponent Properties
Multiplying Fractions
Adding and Subtracting Rational Expressions With the Same Denominator
Quadratic Expressions - Completing Squares
Adding and Subtracting Mixed Numbers with Different Denominators
Solving a Formula for a Given Variable
Factoring Trinomials
Multiplying and Dividing Fractions
Multiplying and Dividing Complex Numbers in Polar Form
Power Equations and their Graphs
Solving Linear Systems of Equations by Substitution
Solving Polynomial Equations by Factoring
Laws of Exponents
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Systems of Linear Equations
Properties of Rational Exponents
Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
Dividing Fractions
Factoring a Polynomial by Finding the GCF
Graphing Linear Equations
Steps in Factoring
Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
Solving Exponential Equations
Finding the GCF of a Set of Monomials
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Graphing Compound Inequalities

We can write compound inequalities with two variables just as we do for one variable. For example,

is a compound inequality. Because the inequalities are connected by the word and, a point is in the solution set to the compound inequality if and only if it is in the solution sets to both of the individual inequalities. So the graph of this compound inequality is the intersection of the solution sets to the individual inequalities.


Example 1

Graphing a compound inequality with and

Graph the compound inequality


We first graph the equations y = x - 3 and . These lines divide the plane into four regions as shown in figure (a) below.

Now test one point of each region to determine which region satisfies the compound inequality. Test the points (3, 3), (0, 0), (4, -5), and (5, 0):

3 > 3 - 3 and 3 Second inequality is incorrect.
0 > 0 - 3   0 Both inequalities are correct.
-5 > 4 - 3   -5 First inequality is incorrect.
0 > 5 - 3   5 Both inequalities are incorrect.

The only point that satisfies both inequalities is (0, 0). So the solution set to the compound inequality consists of all points in the region containing (0, 0). The graph of the compound inequality is shown in figure (b) above.

Compound inequalities are also formed by connecting individual inequalities with the word or. A point satisfies a compound inequality connected by or if and only if it satisfies one or the other or both of the individual inequalities. The graph is the union of the graphs of the individual inequalities.


Example 2

Graphing a compound inequality with or

Graph the compound inequality 2x - 3y 6 or x + 2y 4.


First graph the lines 2x - 3y = -6 and x + 2y = 4. If we graph the lines using x- and y-intercepts, then we do not have to solve the equations for y. The lines are shown in figure (a) below. The graph of the compound inequality is the set of all points that satisfy either one inequality or the other (or both). Test the points (0, 0), (3, 2), (0, 5), and (-3, 2). You should verify that only (0, 0) fails to satisfy at least one of the inequalities. So only the region containing the origin is left unshaded. The graph of the compound inequality is shown in figure (b) below.

Helpful Hint

When graphing a compound inequality connected with“or,” shade the region that satisfies the first inequality and then shade the region that satisfies the second inequality. If the inequalities are connected with “and,” then you must be careful not to shade too much.

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