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
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
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
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
Quadratic Functions
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
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
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Solving Linear Inequalities

Solving a linear inequality means writing it in a form so that only y appears on the left-hand side, and only x and constants appear on the right side. This is similar to solving an equation for y. It is easy to graph the solution sets for linear inequalities. Simply graph the corresponding boundary line and then shade the region either above or below the line.

Solving linear inequalities can be done by using the Addition and Multiplication Properties of Inequalities, just as the corresponding properties of equations are used to solve linear equations. This is best illustrated by examples.

Example 1

Solve y + 7 < 2 x + 5.


First, rewrite the inequality so that y is on the left-hand side by itself.

y + 7 < 2 x + 5 Subtract 7 from each side.
y < 2 x - 2  

So, the solution set is the set of all points that lie below the line y = 2x - 2, as shown in the graph below.

Notice that there is a dashed line on the boundary, since the inequality symbol is < , not .


Example 2

Solve 2y x + 6.


To solve the inequality, first divide each side by 2 to isolate the y .

2y x + 6  
y Divide each side by 2.

The solution set is the set of points on and above the line , as shown in the graph below


Example 3

Solve 3 - y x + 5.


First, use the properties of inequalities to isolate the y.

3 - y x + 5  
- y x + 2 Subtract 3 from each side.
y - x - 2 Multiply each side by -1. Reverse the inequality symbol.

The graph of y x - 2 is shown below.

Notice that a solid line is on the boundary of the region.

Recall that the Multiplication Property of Inequalities states that multiplying each side of an inequality by the same positive number produces an equivalent inequality. However, when multiplying each side by the same negative number, the inequality symbol must be reversed to get an equivalent inequality.

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