Graphing and Writing Linear Functions
Linear Equations and Graphing
Systems of Linear Equations
Solving Polynomial Equations
Matrix Equations and Solving Systems of Linear Equations
Introduction Part II and Solving Equations
Linear Algebra
Graphing Linear Inequalities
Using Augmented Matrices to Solve Systems of Linear Equations
Solving Linear Inequalities
Solution of the Equations
Linear Equations
Annotated Bibliography of Linear Algebra Books
Write Linear Equations in Standard Form
Graphing Linear Inequalities
Introduction to Linear Algebra for Engineers
Solving Quadratic Equations
Systems of Linear Equations
Review for First Order Differential Equations
Systems of Nonlinear Equations & their solutions
Quadratic Equations
Syllabus for Differential Equations and Linear Alg
Linear Equations and Matrices
Solving Linear Equations
Slope-intercept form of the equation
Linear Equations
Linear Equation Problems
Systems of Differential Equations
Linear Algebra Syllabus
Quadratic Equations and Problem Solving
The Slope-Intercept Form of the Equation
Final Exam for Matrices and Linear Equations
Linear Equations
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

• An inequality is an equation with ≤,<,≥ or > instead of an equal sign
• A linear inequality is an inequality that is linear in form (there are no roots, higher powers,
rationals, etc)
• Solving inequalities is the same as solving regular equations, but you need to be careful of the
direction of the inequality sign. Multiplying or dividing the equation by a negative swaps the
direction of the inequality sign (i.e. > would become <)
• A compound inequality is formed by putting together two inequalities with the word
− and (meaning intersection, both have to be satisfied to be a solution)
− or (meaning union, either has to be satisfied to be a solution)
• The compound inequality with and could be joined in one statement called a conjunction in the
manner below
x > 5 and x ≤12 5 < x ≤12
• The compound inequality with or is called a disjunction and cannot be rewritten in shorter form

Recap of the Properties:
• Adding any quantity to both sides won’t change the inequality
i.e. a < b and k is any real, =>a + k < b + k
• Multiplying both sides by a POSITIVE constant won’t change the inequality
i.e. a < b, k > 0=>ak < bk
• Multiplying both sides by a NEGATIVE constant WILL change the direction of the inequality
i.e. a < b, k < 0=>ak > bk
• Taking the reciprocal WILL change the direction of the inequality

Describing the Intervals:
• There are several ways to describe intervals
– interval notation uses parenthesis, “(” and brackets, “]”
– set notation uses inequality symbols, “>”
– graphing uses the number line with open and closed circles connected with lines or shading
• An interval can be open (meaning you don’t actually “contain” the endpoint). The open end is
described with “)” “>” and/or open circles
• Or an interval can be closed (meaning you do contain the endpoint). The closed end is described
with “]” “ ≥ ” and/or closed circles
• Example. To describe the values for x greater than 5
x > 5

Some Common Mistakes (with NON-linear equations):
• When solving for a product of two quantities, you must take into account that a negative times a
negative is a positive.
i.e. to solve for (x − 3)(x + 2) < 0 , you can’t just take when both are negative. You have to do
what is called a “sign diagram” This can be done in one of two ways…
1) Look at the sign of x – 3 and x + 2 separately on a chart

You can see that (x − 3)(x + 2) < 0 when –2 < x < 3

2) Still do a chart, but just “test” values to the left and right of your “zeros”
To the left of –2 (say –4), (−4 − 3)(−4 + 2) > 0
Between –2 and 3 (say 0), (0 − 3)(0 + 2) < 0
To the right of 3 (say +4), (4 − 3)(4 + 2) > 0

Which gives the same result as above

• Don’t ever just divide by x if you’re trying to solve an inequality for x. Not only are you neglecting
values, you may be inadvertently dividing by a negative.
i.e. x2 < x does not reduce to x <1.
Solving correctly you get x2 − x < 0, x(x −1) < 0 so (with sign diagram) 0 < x < 1

• When solving inequalities that contain a squared term, you can’t just take the square root as
normal. You have to include the +/- with the variable, NOT the constant term.
i.e. to solve

Some Exercise Problems:
• Example. Solve 5y −5+ y ≤ 2 − 6y −8

• Example. Solve (x +1)(x + 2) > x(x +1)

• Example. Solve −3 <1− 2x ≤ 3

• Example. Solve 5x +11≤ −4 or 5x +11≥ 4

Absolute Value with Inequalities:
in other words, − k ≤ x ≤ k

• Example. Solve