MAT.116 1.8-1.9

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MAT.116 1.8-1.9

1.9 Inequalities and Absolute Value

Absolute Value

If a and b are any real numbers, then

Definition

The absolute value of a number a is denoted by |a| and is defined by

|a| = { a if a>=0

{ -a if a < 0

Absolute value represents the distance a real number is from 0 on the number line. Absolute value is always nonnegative.

Inequalities

Polynomial and Rational Inequalities

A graphing calculator is a good tool to assist with the solution of nonlinear inequalities.

Solving

A real number is a solution of an inequality in one variable if a true statement is obtained when the variable is replaced by that number.

The set of all real numbers satisfying the inequality is called the solution set.

Solution sets for inequalities in one variable can be written in inequality notation, set notation, and interval notation, and they can be graphed on a number line

Properties

Let a, b, and c be any real numbers.

These properties are also true for >, >=, and <=.

Symbols

< less than (strictly less than)

> greater than (strictly greater than)

<= less than or equal to

>= greater than or equal to

Equivalent symbols:

Intervals

Notation

Finite

Infinite

Vocabulary

Finite intervals

Open intervals

Closed intervals

Half-open intervals

Infinite intervals

1.8 Quadratic Equations

Solutions

The disciriminant of the quadratic equation is

b^2 - 4ac.

If the discriminant is positive, the equation has two distinct real solutions.

If the discriminant is zero, the equation has one real solution.

If the disciminant is negative, the equation has no real solutions.

Using the Quadratic Formula

Given a quadratic equation in standar form,

ax^2 + bx + c = 0 (a not zero),

we can solve the equation using the quadratic formula

-b +- sqrt( b^2 - 4ac )

x = -------------------------

2a

Solving by Completing the Square

Begin with the quadratic equation in standard form

ax^2 + bx + c = 0

Rewrite the equation into the equivalent form

x^2 + (b/a)x = -(c/a)

Add (b/2a)^2 to both sides of the equation

x^2 + (b/a)x + (b/2a)^2 = -(c/a) + (b/2a)^2

Factor the left side into a perfect square and add the right-hand side togther

(x + b/2a)^2 = (b^2 - 4ac) / 4a^2

Use the square root principle and solve for x

x + b/2a = +-sqrt((b^2 - 4ac) / 4a^2)

x = -b/2a +- sqrt((b^2 - 4ac) / 4a^2)

Solving by Factoring

Zero-Product Property of Real Numbers:

If a and b are real numbers and ab=0, then a=0, or b=0, or both a,b=0.

Quadratic Equations

A quadratic equation in one variable can be written in the form ax^2+bx+c=0 (a not zero). This is called standard form.