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!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# 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 â€” 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:

â€¢ Points above the line.

â€¢ Points on the line.

â€¢ 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 + 6Is 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

â€¢ (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.

â€¢ (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.

â€¢ (-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.