Learning Objectives for Section 1.1 Linear Equations and Inequalities

   The student will be able to solve linear equations.

The student will be able to solve linear inequalities.

The student will be able to solve applications involving linear equations and inequalities.

Linear Equations, Standard Form

In general, a

LINEAR EQUATION

in one variable is any equation that can be written in the form

ax



b

 0 where

a

is not equal to zero. A linear equation in one variable is also called a

FIRST DEGREE EQUATION

. The greatest degree of the variable is 1.

Linear Equations, Standard Form

For example, the equation 3  2 (

x

 3 ) 

x

3  5 is a linear equation because it can be converted to standard form by clearing of fractions and simplifying.

Which are linear equations?

1) 5

x

2 2)  0.75

x

 12 3) 5(3

x

21 2

x

 7 4)

x x

4

Equivalent Equations

Two equations are

equivalent

if one can be transformed into the other by performing a series of operations which are one of two types: 1. The same quantity is ___________ to or ____________ from each side of a given equation.

2. Each side of a given equation is _____________ by or _____________ by the same nonzero quantity.

To

solve

a linear equation, we perform these operations on the equation to obtain simpler equivalent forms, until we obtain an equation with an obvious solution.

Example of Solving a Linear Equation Example:

Solve

x

 2 2 

x

3  5

Formulas

A formula is an equation that relates two or more variables.

Some examples of common formulas are: 1.

A = πr 2 2.

I = Prt

3.

y

= m

x

+ b 7

Solving a Formula for a Particular Variable Example:

5

x

 3

y

 12

y

.

Solving a Formula for a Particular Variable Example:

C

 5 9 

F

 

F

.

Solving a Formula for a Particular Variable Example:

Solve

M=Nt+Nr

for

N

.

Linear Inequalities

If the equality symbol = in a linear equation is replaced by an inequality symbol (<, >, ≤, or ≥), the resulting expression is called a

first-degree inequality or linear inequality

. For example, 5 

x

 2  2

x

is a linear inequality.

Solving Linear Inequalities

We can perform the same operations on inequalities that we perform on equations, EXCEPT THAT …………………….

THE DIRECTION OF THE INEQUALITY SYMBOL REVERSES IF WE MULTIPLY OR DIVIDE BOTH SIDES BY A NEGATIVE NUMBER.

Solving Linear Inequalities

•For example, if we start with the true statement -2 > -9 and multiply both sides by 3 , we obtain:

The direction of the inequality symbol remains the same.

------------------------------------------- •However, if we multiply both sides by

-3

instead, we must write to have a true statement.

The direction of the inequality symbol reverses.

Example for Solving a Linear Inequality

Solve the inequality 3(

x

-1) < 5(

x

+ 2) - 5

Interval and Inequality Notation

If

a is less than b

, the

double inequality a < x < b

means that

a < x

and

x < b

. That is,

x is between a and b

.

Example

Solve the double inequality: 2 3

t

Interval Notation INTERVAL NOTATION

is also used to describe sets defined by single or double inequalities, as shown in the following table.

Inequality

a ≤ x ≤ b a ≤ x < b a < x ≤ b a < x < b x ≤ a x < a x ≥ b x > b

Interval

[

a,b

] [

a,b

) (

a,b

] (

a,b

) (-∞

,a

] (-∞

,a

) [

b,

∞) (

b,

∞)

Graph

Interval and Inequality Notation and Line Graphs

(A)Write [-5, 2) as a double inequality and graph .

(B) Write

x

≥ -2 in interval notation and graph.

Interval and Inequality Notation and Line Graphs

 13 (C) Write in interval notation and graph.

2 

x

(D) Write -4.6 <

x

≤ 0.8 in interval notation and graph.

Procedure for Solving Word Problems

1.

2.

3.

4.

5.

Read the problem carefully and

introduce a variable

to represent an unknown

quantity

in the problem.

Identify other quantities

in the problem (known or unknown) and express unknown quantities in terms of the variable you introduced in the first step.

Write a verbal statement using the conditions stated in the problem and then

write an equivalent mathematical statement

(equation or inequality.)

Solve

the equation or inequality and answer the questions posed in the problem.

Check

that the solution solves the original problem.

Example: Break-Even Analysis

A recording company produces compact disk (CDs).

One-time fixed costs for a particular CD are $24,000

; this includes costs such as recording, album design, and promotion.

Variable costs amount to $6.20 per CD

and include the manufacturing, distribution, and royalty costs for each disk actually manufactured and sold to a retailer.

The CD is sold to retail outlets at $8.70 each

. How many CDs must be manufactured and sold for the company to

break even

?

20

Understanding the Vocabulary Total Cost: Revenue: Break-Even Point: Profit:

21

Break-Even Analysis (continued) Solution Step 1.

Define the variable. (Be sure it represents a quantity and always include the appropriate units.) Let

x

=

Step 2.

Identify other quantities in the problem.

Break-Even Analysis (continued) Step 3.

Set up an equation using the variable you defined.

Step 4.

Solve for the variable and answer the question(s) posed. ( Always write out the answer in sentence form, using appropriate units.)

Break-Even Analysis (continued) Step 5.

Check:

Break-Even Analysis (continued) Related Questions:



What is the total cost of producing the CDs at the break-even point?



What is the revenue made at the break-even point?



What is the profit made at the break-even point?



How many CDs must the company make and sell in order to make a profit?

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