Solving Inequalities

1
Solving Inequalities

• Sections 1.2 1.3

2
Inequalities

• An equation states that two numbers (or
  expressions) are the same.
• An inequality says that one number (or
  expression) is larger than the other.
• An equation usually has one (or a few) solutions.
• The solution to an inequality is generally an
  interval of numbers (infinitely many numbers).

3
Rules for Inequalities(hold for lt, lt, gt, gt)

• If a lt b then a c lt b c for any number c.
• If a lt b then ac lt bc for any positive number c.
• If a lt b and c lt 0, then ac gt bc.
• If a lt b and a c then c lt b.
• If a and b are either both positive or both
  negative, and a lt b, then 1/a gt 1/b.
• If a and b are positive then for any positive
  power n, if a lt b then an lt bn.

4
Linear Inequality

• Definition
• A linear inequality in the variable x is an
  inequality that can be written in the form
• ax b lt 0
• where a and b are constants with a ? 0.

5
Solving Linear Inequalities

• Our goal is to get a simple inequality with only
  the variable and a number.
• Solve 4x - 5 lt 6(4 x)
• Multiply out all expressions
• 4x - 5 lt 24 6x
• Add 6x to both sides
• -2x - 5 lt 24
• Add 5 to both sides
• -2x lt 29
• Multiply both sides by -1/2
• x gt -14.5

6
Representing solutions to inequalitiesInterval
notation

• An interval may be closed (includes the
  endpoints), open (endpoints excluded), or
  half-open (only one endpoint included).
• We use a parentheses () to denote an open
  interval and square brackets to denote a
  closed interval.
• In additon an interval may be finite (definite
  endpoints), half-infinite (extends indefinitely
  in one direction), or infinite (extends
  indefinitely in both directions).
• N.B. Parentheses are always used with 8 or - 8.

7
Interval Notation, cont.

• Inequality Number line Interval not.
• a lt x lt b (a, b)

a lt x lt b a, b
a lt x lt b (a, b
a lt x (a, 8)
x lt b (-8, b
8
Applications of Inequalities

• We approach applications involving inequalities
  similar to applications involving equations.
• Assign a variable to the unknown quantity.
• Express any other unknown quantities in terms of
  the variable.
• Write an inequality expressing the given
  information.
• Use the appropriate techniques to solve the
  inequality.
• Interpret the result in terms of the original
  situation.

9
Inequalities - key words

• Phrase Inequality
• more than gt
• at least gt
• no more than lt
• maximum amount lt
• least number gt

10
The Buckeye Widget Company sells their basic
widgets for 2.50 each. The manufacturing cost
per widget is 1.70. If the companys fixed
monthly costs are 6000, what is the least number
of widgets they can produce per month to make a
profit?

• Let n be the number of widgets manufactured.
• Then 2.5n is the total monthly revenue.
• 6000 1.7n is the total monthly cost
• 2.5n - (6000 1.7n) gt 0
• 2.5n - 6000 - 1.7n gt 0
• .8n -6000 gt 0
• .8n gt 6000
• n gt 7500
• 5. They must manufacture at least 7501 widgets to
  make a profit.

11
The Buckeye Widget Company plans to add a new
line of deluxe widgets. These will sell for
4.50 and cost 2.80 to produce. If the total
monthly production capacity is 10,000 widgets
(both deluxe and basic), with the same fixed
costs of 6000, what is the least number of
deluxe widgets they should produce to make more
than 5000 monthly profit?

• Let d be the number of deluxe widgets produced.
• Then 10,000 - d is the number of basic widgets
  produced.
• Total monthly revenue is 4.5d 2.5(10,000 - d).
• Total monthly cost is 6000 2.8d 1.7(10,000 -
  d).
• 4.5d 2.5(10,000 - d) - (6000 2.8d
  1.7(10,000 - d)) gt 5000
• 4.5d - 2.5d -2.8d 1.7d 25000 - 6000 - 17000 gt
  5000
• .9d -2000 gt 5000
• .9d gt 3000
• d gt 3333 1/3
• 5. They should manufacture at least 3004 deluxe
  widgets monthly.

12
A construction company has contracted with the
state to build a new freeway interchange. The
state has projected 6 months for the completion
of the project and offered the company two
payment options. Option 1 is a fixed sum of
2,000,000 if they finish in 6 months. Option 2
offers 1,000,000 plus 10,000 for every day less
than 183 they take to finish the project. How
long should they take to complete the project if
they choose option 2?

• Let d be the number of days to complete the
  project.
• 1,000,000 10,000(183 - d) is the total pay
  under option 2
• 1,000,000 10,000(183 - d) gt 2,000,000
• 1,000,000 1,830,000 - 10,000d gt 2,000,000
• -10,000d gt -830,000
• d lt 83
• 5. The company should finish the project in 82
  days or less if they choose option 2.

13
Key Suggested Problems

• Sec. 1.2 5, 9, 13, 19, 21, 27, 35
• Sec. 1.3 1, 3, 7, 11, 13
• www.math.ohio-state.edu/hambrock/130/