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Fields Medal Prize Winners (1998)




TUTORIALS:


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
Fractions
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
Roots
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
Powers
Rewriting Algebraic Fractions
Exponents
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 Linear Inequalities

Use the following procedure to graph a linear inequality in two variables.

Procedure — To Graph a Linear Inequality in Two Variables

Step 1 Graph the equation that corresponds to the given inequality.

• If the inequality symbol is or , use a solid line to show that points on the line are solutions of the inequality.

• If the inequality symbol is < or >, use a dotted line to show that points on the line are not solutions of the inequality.

Step 2 Use a test point NOT on the line to determine the region whose points satisfy the inequality.

Step 3 Shade the region whose points satisfy the inequality.

 

Example 1

Graph the inequality 2x - y < 4.

Solution

Step 1 Graph the equation that corresponds to the given inequality.

Graph the equation 2x - y = 4.

To do this, substitute 0 for y and solve for x to get the x-intercept, (2, 0).

Next, substitute 0 for x and solve for y to get the y-intercept, (0, -4).

Then, plot the points.

Since the inequality symbol “<” does not contain “equal to,” draw a dotted line through the plotted points.

The dotted line shows that points on the line are not solutions of the inequality.

Step 2 Use a test point NOT on the line to determine the region whose points satisfy the inequality.

The point (0, 0) is not on the line, so it can be used as a test point.

Substitute 0 for x and 0 for y.

Simplify.

Is

Is

2(0) - 0

0

< 4 ?

< 4 ? Yes

Since 0 < 4 is true, the ordered pair (0, 0) is a solution of the inequality 2x - y < 4.

This means all the points in the region containing (0, 0) are solutions.

Step 3 Shade the region whose points satisfy the inequality.

Shade the region that includes (0, 0). This is the region above the dotted line.

Note — Using (0, 0) as a Test Point

If the point (0, 0) does not lie on the line, it is a good test point since it often makes the calculations easier.