FreeAlgebra
Tutorials!
Home
Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Powers
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
http:
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Fractions
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Adding
Quadratic Functions
Conjugates
Factoring
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Percents
Arithmetics
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Polynomials
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
http:
Solving Inequalities with Fractions and Parentheses
http:
Multiplying Polynomials
Fractions
Solving Quadratic and Polynomial Equations
Extraneous Solutions
Fractions
 
Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Solving Linear Inequalities

Suppose we have two expressions, 2x + 3 and -2y + 9.

A linear equation, such as 2x + 3 = -2y + 9, consists of two expressions joined by an equals sign, =.

A linear inequality, such as 2x + 3 < -2y + 9, consists of two expressions joined by an inequality symbol: <, , , or >.

Definition — Linear Inequality in Two Variables

A linear inequality in two variables, x and y, is an inequality that can be written in one of the following forms:

Ax + By < C, Ax + By C, Ax + By > C, Ax + By C, where A, B, and C are real numbers and A and B are not both 0.

A solution of a linear inequality in two variables is an ordered pair, which when substituted in the inequality, results in a true statement. That is, a solution that satisfies the inequality.

A linear inequality, such x + y < 3, has infinitely many solutions. To identify these solutions, we often use the graph of the corresponding linear equation.

For example, consider the graph of the linear equation x + y = 3.

Notice that the line divides the xy-plane into three regions:

• Points above the line.

• Points on the line.

• Points below the line.

Points on the line, such as (-3, 6), are solutions of the equation x + y = 3.

Points NOT on the line are solutions of one of the following inequalities:

x + y < 3

x + y > 3.

Example 1

Determine if each ordered pair is a solution of x + y < 3.

a. (0, 0)

b. (5, 0)

c. (-3, 6)

Solution

a. Substitute 0 for x and 0 for y.

Simplify.

Is 0 + 0

Is 0

< 3 ?

< 3 ?  Yes

  Since 0 < 3 is true, the ordered pair (0, 0) is a solution of x + y < 3.

Notice that the point (0, 0) lies below the line x + y = 3.

b.  Substitute 5 for x and 0 for y.

Simplify.

Is 5 + 0

 Is 5

< 3 ?

< 3 ? No

  Since 5 < 3 is false, the ordered pair (5, 0) is NOT a solution of x + y < 3.

Notice that the point (5, 0) lies above the line x + y = 3.

c. Substitute -3 for x and 6 for y. 

Simplify. 

Is -3 + 6

Is 3

< 3 ?

< 3 ? No

  Since 3 < 3 is false, the ordered pair (-3, 6) is NOT a solution of x + y < 3.

Notice that the point (-3, 6) lies on the line x + y = 3.

Recall that a line divides the xy-plane into three regions: points on the line and points on either side of the line.

If a point in a region satisfies an inequality, then every point in that region satisfies the inequality. Likewise, if a point in a region does not satisfy an inequality then no point in that region satisfies the inequality.

In the previous example we found that

• (0, 0) is a solution of x + y < 3.

Since (0, 0) lies below the line x + y = 3, every point below the line a solution of x + y < 3.

To graph these solutions, shade the region below the line.

• (5, 0) is not a solution of x + y = 3.

Since (5, 0) lies above the line x + y = 3, no point above the line is a solution of x + y < 3.

• (-3, 6) is not a solution of x + y < 3.

Since (-3, 6) lies on the line x + y = 3, no point on the line is a solution of x + y < 3.

To show this, we use a dotted line for x + y = 3.

The points (0, 0) and (5, 0) are often called test points.

A test point is a point that we substitute in a linear inequality to determine the region of the xy-plane that represents the solution of the inequality.

The solution of the inequality x + y < 3 is the set of all ordered pairs in the region below the line x + y = 3. The graph of the inequality is the shaded region. (The line is not included.)

Note:

For a test point, we can use any point NOT on the line.

You may want to choose a test point from each side of the line to check that you have found the region that represents the solution.

All Right Reserved. Copyright 2005-2007