Fields Medal Prize Winners (1998)


Adding and Subtracting Monomials
Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Rewriting Algebraic Fractions
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Graphing Compound Inequalities

Graphing the Solution Set to a Compound Inequality

The following example shows a compound inequality that has no solution and one that is satisfied by every real number.

Example 1

All or nothing

Sketch the graph and write the solution set in interval notation to each compound inequality.

a) x < 2 and x > 6

b) x < 3 or x > 1


a) A number satisfies x < 2 and x > 6 if it is both less than 2 and greater than 6. There are no such numbers. The solution set is the empty set, Ø.

b) To graph x < 3 or x > 1, we shade both regions on the same number line as shown in the figure below. Since the two regions cover the entire line, the solution set is the set of all real numbers (-, ).


If we start with a more complicated compound inequality, we first simplify each part of the compound inequality and then find the union or intersection.


Example 2


Solve x + 2 > 3 and x - 6 < 7. Graph the solution set.


First simplify each simple inequality:

x + 2 - 2 > 3 - 2 and x - 6 + 6 < 7 + 6
 x > 1 and  x < 13

The intersection of these two solution sets is the set of numbers between (but not including) 1 and 13. Its graph is shown in the figure below. The solution set is written in interval notation as (1, 13).


Example 3


Graph the solution set to the inequality

5 - 7x 12 or 3x - 2 < 7.


First solve each of the simple inequalities:

5 - 7x - 5 12 - 5 or 3x - 2 + 2 < 7 + 2
-7x 7 or 3x  < 9
x -1 or x < 3

 The union of the two solution intervals is (-, 3). The graph is shown in the figure below.