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Systems of Linear Equations Introduction
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Solving 2nd Degree Equations
Review Solving Quadratic Equations
System of Equations
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Linear Equations Functions Zeros, and Applications
Rational Expressions and Functions
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Lesson Plan for Comparing and Ordering Rational Numbers
LinearEquations
Solving Equations
Radicals and Rational Exponents
Solving Linear Equations
Systems of Linear Equations
Solving Exponential and Logarithmic Equations
Solving Systems of Linear Equations
DISTANCE,CIRCLES,AND QUADRATIC EQUATIONS
Solving Quadratic Equations
Quadratic and Rational Inequalit
Applications of Systems of Linear Equations in Two Variables
Systems of Linear Equations
Test Description for RATIONAL EX
Exponential and Logarithmic Equations
Systems of Linear Equations: Cramer's Rule
Introduction to Systems of Linear Equations
Literal Equations & Formula
Equations and Inequalities with Absolute Value
Rational Expressions
SOLVING LINEAR AND QUADRATIC EQUATIONS
Steepest Descent for Solving Linear Equations
The Quadratic Equation
Linear equations in two variables
   
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Quadratic and Rational Inequalities

A quadratic inequality is any
inequality that can be put in
one of the forms

where a, b, and c are real numbers and a≠0.

Solving Quadratic Inequalities

Step 1: write the inequality in standard form.
Step 2: solve the related quadratic equation.
Step 3: locate the boundary points on a
number line.
Step 4: construct a sign chart.
Step 5: the solution set is the interval that
produced a true statement.

Example 1: Solve a Quadratic
Inequality


Solve:

Solution:
Step 1: Write the inequality in
standard form.



Step 2: Solve the related quadratic equation.

(x-1)(x-3)=0

x+1=0 or x-3=0

x=-1 or x= 3

The boundary points are -1 and 3.

These two points divide the number
line into three test intervals, namely
(−∞,-1) ,(−1,3) ,and (3,∞) .

Take a test point within each
interval and check the sign.

Test
Interval
Test
Point
X+1 X-3 (x+1)(x-3)
(−∞,-1) -2 - - +> 0
(−1,3) 0 + + -< 0
(3,∞) 4 + + +>0


The question is "where ?"

Our table shows that


in the interval (-1,3).

So the solution set is
the interval (-1,3)

Practice Exercises

Solve: Answers:
(1,3)

All real

Solving Rational Inequalities

Solve:

It is incorrect to multiply both sides by
x-2 to clear fractions. The problem is
that x-2 contains a variable and can be
positive or negative, depending on the
value of x. Thus, we do not know
whether or not to reverse the sense of
inequality.

Example 2:

solution: We begin by finding values
of that make the numerator and
denominator 0

Set the numerator and denominator equal to 0.

Solve

 x=-5 and x=2

The boundary points are -5 and 2.

Locate boundary points -5 and 2
on a number line.

These boundary points divide the
number line into three intervals,
namely (−∞,-5),(-5,2), and (2,∞).
Now, construct a sign chart: take
one test point from each interval
and check the signs.

Intervals Test
Points
x+5 x-2
(−∞,-5) -6 - - +> 0
(-5,2) 0 + - -< 0
(2,∞) 3 + + +>0

The given question is "for what values
of the quantity is positive?"

Our table shows that for

(−∞,-5) or (2,∞).

Thus the solution set is

(−∞,-5) or (2,∞).

The graph of the solution set on a
number line is shown as follows:

Practice Exercises

Solve: Answers:
(-5,-2)

(−∞,3) or [4,∞)

[-4,-2]
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