Systems of Equations
Adding and Subtracting Rational Expressions with Different Denominators
Graphing Linear Equations
Raising an Exponential Expression to a Power
Horizontal Line Test
Quadratic Equations
Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
Factoring Polynomials by Grouping
Solving Linear Systems of Equations
Exponential Functions
Factoring Trinomials by Grouping
The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
Solving Absolute Value Equations
Solving Right Triangles
Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
Multiplying Polynomials
Slope of a Line
Multiplying Rational Expressions
Percent of Change
Equations Involving Fractions or Decimals
Simplifying Expressions Containing only Monomials
Solving Inequalities
Quadratic Equations with Imaginary Solutions
Reducing Fractions to Lowest Terms
Prime and Composite Numbers
Dividing with Exponents
Dividing Rational Expressions
Equivalent Fractions
Graphing Quadratic Functions
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
Solving Quadratic Equations
Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
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Solving Linear Inequalities in One Variable


Example 1:

Solve 6x − 3 < 2x + 5

1) Add opps:

Complete the step:


Now (6-2) = 4 and (5+3) = 8

4 is the coefficient of x

2) Multiply recip:


Since and

 Note that 1 < 2 so use x = 1 as a replacement for x to check your answer.




Replace x with 1 and simplify.


x < 2 is correct.

Always substitute a number in the result for x in the original inequation and check to see if that makes a “true statement”. Then you know that your result is the solution set or “answer” to the inequality.

[NOTE: For inequations there are many solutions in the solution set.]

We chose x = 1 as a replacement value to check the algebraic inequality, but we could have chosen any value on the REAL number line to the left of 2 even x = 1.999999999 which is less than 2.

Since we can represent any REAL number on the number line we can represent ALL of the numbers that can be chosen as a replacement for the variable by shading or drawing a bold line with an arrow pointing in the direction of all other real numbers. Since x ≠ 2 we left a hole at that spot on the number line.

You should always sketch the solution to inequalities on the number line as an aid to choosing replacement values.

There are times when we want to know all of the values that are “at most 2” . From the table at the beginning of this session we see that the symbol for “at most” is ≤ which is a compound symbol.

This is really two problems in one: an equation and inequality which has the same point on the number line as a boundary. In the first case 2 is a replacement value for x and in the second case the numbers up to, but not including, 2 are replacement values. To show both cases on the number line we “fill in the hole” or put a large “dot” on that spot to represent that 2 also belongs to the replacement set.

Similarly, we can find the “solutions” or replacement values for “at least 5” which we see can be represented by the symbol ≥ and can be shown on the number line by a bold arrow pointing to the right and a “dot” to fill in the hole.


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