Transcript faculty.clintoncc.suny.edu

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.
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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 FIRSTDEGREE EQUATION. The greatest degree of the
variable is 1.
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Linear Equations, Standard Form
For example, the equation
3  2 ( x  3) 
x
5
3
is a linear equation because it can be converted to standard
form by clearing of fractions and simplifying.
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Which are linear equations?
1) 5 x  1  21
2
2)  0.75 x  12
3) 5(3 x  1)  2 x  7
4)
x3  x4
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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.
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Example of Solving a
Linear Equation
Example: Solve
x2
2

x
5
3
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Formulas
A formula is an equation that relates two or more variables.
Some examples of common formulas are:
1. A = πr2
2. I = Prt
3. y = mx + b
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Solving a Formula for a
Particular Variable
Example: Solve 5 x  3 y  1 2
for y.
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Solving a Formula for a
Particular Variable
Example: Solve C 
5
9
F
 32 
for F.
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Solving a Formula for a
Particular Variable
Example: Solve M=Nt+Nr for N.
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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  1  3 x  2 
x
2
is a linear inequality.
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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.
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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.
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Example for Solving a
Linear Inequality
Solve the inequality
3(x-1) < 5(x + 2) - 5
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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:
1 
2
t  5  11
3
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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
Interval
[a,b]
[a,b)
a<x≤b
a<x<b
x≤a
(a,b]
(a,b)
(-∞,a]
x<a
x≥b
x>b
(-∞,a)
[b,∞)
(b,∞)
Graph
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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.
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Interval and Inequality Notation
and Line Graphs
(C) Write

13
2
 x
in interval notation and graph.
(D) Write -4.6 < x ≤ 0.8 in interval notation and graph.
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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.
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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?
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Understanding the Vocabulary
Total Cost:
Revenue:
Break-Even Point:
Profit:
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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.
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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.)
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Break-Even Analysis
(continued)
Step 5. Check:
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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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Examples from Text
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Examples from Text
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