Download Solving Linear and Quadratic Equations and Inequalities and more Slides Algebra in PDF only on Docsity!
1.1 Linear and Rational Equations
In this section you will learn to:
recognize equations that are identities, conditional, or contradictions
solve linear equations in one variable
find restrictions on variable values
solve rational equations with variables in the denominator
solve formulas for a specific value
Types of Equations
Identity Conditional Contradiction/Inconsistent
Solution set consists of all
real numbers*.
Solution set consists of one or
more solutions (but not an
identity).
The equation has no solutions.
Example: x 2 5 x 3
2 = 2 (or 0 = 0)
(Both sides are “identical”.)
Example: 5 6 0
2 x x
x = 2 or x = 3
Example: x 2 6 x 3
(Since 2 3 , there is no solution.)
ALWAYS TRUE SOMETIMES TRUE NEVER TRUE
*except when one or both sides of the equation results in division by 0 (Example: 2
x
x
)
Solving Linear or First-Degree Equations ( ax b 0 , where a and b are real numbers and a 0 )
Example 1: 5 x ( 2 x 2 )x( 3 x 5 ) Steps: (if coefficients are integers)
- Simplify each side of the equation.
- Move variable terms to one side and
constant terms to the other.
- Isolate the variable. (Multiply or divide
based on the coefficient.)
- Check the solution in the original equation.
When “solving” an equation, be sure to write
your answer in solution set form.
Solution Set:
Type of Equation: ___________________________
Example 2: Solve 3( t 5) 6 t 5 3(1 3 )t
Solution Set:
Type of Equation: _____________________________
Example 3: Solve: 2
3 x
x x
(Hint: If coefficients are fractions, first multiply
sides of the equation by the Lowest Common
Denominator, or LCD, to clear the fractions.)
Solution Set:
Type of Equation: ________________________
A rational expression is the quotient of two polynomials as shown in Example 4 below. The domain
of a rational expression is the set of real numbers for which the expression is defined. Since division by
zero is not defined, numbers that make the denominator zero must be excluded from the domain.
A rational equation is an equation that contains rational expressions. When solving this type of
equation, you may need to multiply both sides by a quantity (generally the LCD) which may contain a
variable. (When the variable appears in the denominator the equation is no longer a linear equation.)
However, you must exclude any values of the variable that make the denominator zero as this would
give a false solution, called an extraneous solution.
Example 4: Find all real numbers that must be excluded from the domain of each rational expression.
a
a
2 z 5
2
m 3 m 4
t q 5
q
Example 8: Solve: 2
b
b b b
What value(s) of b must be excluded? _________
Type of Equation: ________________________
Solution Set:
Solving Formulas for a Specific Value (or in terms of another variable):
Example 9: Solve (^) V Bh
for^ B.^ Steps:
- If fractions are involved, multiply by LCD.
- Move terms with desired variable to one
side. Move all other terms to other side.
- If two or more terms contain the desired
variable, FACTOR out the variable.
- Divide both sides by the “non variable”
factor.
Example 10: Solve S = P + Prt for t.
When solving for t, Sam’s answer was
Pr
S P
t
while Sophia had
Pr
S
t
r
for an answer. Which
student had the correct answer? Explain.
Example 11: Solve S = P + Prt for P.
This time Sam’s answer was
S
P
rt
and Sophia simplified her answer as
S
P S
rt
. Which student
had the correct answer. Explain.
Example 12: Solve
c
a b
x
for b. Example 13: Solve
A B
A B
C
for A.
Example 14: A formula for converting temperature from degrees Fahrenheit to degrees Celsius is
C F Use this formula to convert the temperatures below.
95 F _______ C
25 C _______ F
Now solve the formula for F (Fahrenheit in terms of Celsius) and then convert the given temperatures.
25 C _______ F
0 C _______ F (freezing pt. for water)
100 C _______ F(boiling pt. for water)
1.1 Homework Problems
- Find all real numbers that must be excluded from the domain of each rational expression.
(a) 2
x 4
(b)
x
x x
(c) 3
x
x x
(d) 2
x 3 x 4
For Problems 2–8, solve the equation and then determine if it is an example of an identity,
contradiction or conditional.
- 2 x 4 ( 5 x 1 ) 3 x 38 3.
x x
x x
- 14 x 10 11 (x 3 ) 3 x 6.
x x x 2 x
2
- 7 x 21 7 (x 3 )
2
x x x
x
- Solve C 2 r for r. 10. Solve a 2 b 3 c for c.
- Solve 3
b
y
x
a
for x. 12. Solve
A B
A B
C
for B.
- Solve
D
AC D
A B
for D. 14. Solve:
a b c
for b.
- Solve 5
x y
for x. 16. Solve
z
w
x y
for z.
- An example of a formula used to model the proportion of correct responses in terms of the number of
trials of a particular task is
x
P
x
, where P is the proportion of correct responses after x
trials. How many learning trials are necessary for 0.95 of the responses to be correct?
1.1 Homework Answers: 1 .(a) x 2 (b) x 4, 0 (c) x 0 , 3 (d) x 4, 1 2. {-2}; conditional
- {2}; conditional 4.
; conditional 5. ; contradiction 6.
; conditional
- all real numbers; identity 8. {2}; conditional 9.
C
r 10.
2 b a
c
b y
ab
x
C
AC A
B 13.
A B
AC
D 14.
c a
ac
b
y
x
y
2 w
z
x y
- About 11 learning trials
1.2 Applications of Linear Equations
In this section you will learn to:
use linear equations to solve word problems
Example 1: The number of cats in the U. S. exceeds
the number of dogs by 7.5 million. The number of
cats and dogs combined is 114.7 million. Let c denote
the number of cats. Write an equation in terms of the
variable c which models this information. Then use
your equation to find the number of cats and dogs.
Example 2: After a 30% discount, a cell phone
sells for $224. Find the original price of the cell
phone before the discount was applied to the
purchase.
Example 3: Including a 6% sales tax, an item
costs $91.69. Find the cost of the item before
the sales tax was added.
Steps/Tips for Solving Word Problems:
- Read the problem carefully. Underline key words and phrases. Let x (or any variable)
represent one of the unknown quantities. Draw a picture or diagram if possible.
- If necessary write expressions for other unknowns in terms of x.
- Write a verbal model of the problem and then replace words with numbers, variables
and/or symbols.
- Solve the equation. Answer the question in the problem. Label answers!
- Check your answer(s) in the original word problem (not in your equation).
Example 8: How many liters of a 9% solution of salt
should be added to a 16% solution in order to obtain
350 liters of a 12% solution?
Example 9: A student scores 82%, 86% and 78% on her
first three exams. What score is needed on the fourth exam
for the student to have an average of 85% for all four exams?
Example 10: A student scores 85%, 72%, 96%, and 98% for
his first four chapter exams. If the fifth exam, the final exam,
counts twice as much as each of the chapter exams, is it possible
for the student to get a high enough final exam score to get a
90% average for the course?
Example 11: A rectangular swimming pool measures
18 feet by 30 feet and is surrounded by a path of uniform
width around all four sides. The perimeter of the rectangle
formed by the pool and the surrounding path is 132 feet.
Determine the width of the path.
Example 12: A Delta jet leaves Lansing International Airport
and travels due east at a rate of 500 mph. A United jet takes
off 15 minutes later traveling in the same direction at 650 mph.
How long will it take for the United jet to catch up to
the Delta jet? (Be sure to use consistent units for time!)
d = r t
Delta
United
Example 13: Sam can plow a parking lot in 45 minutes.
Eric can plow the same parking lot in 30 minutes. If Sam
and Eric work together, how long will it take them to clear
the entire parking lot? First complete the table to find what
fraction of the job can be completed in the given number of
minutes for each worker and then write your equation.
1 minute 2 minutes 5 minutes n minutes
Eric
Sam
1.3 Quadratic Equations
In this section you will learn to:
solve quadratics equations by
- factoring
- square root property
- quadratic formula
- graphing (used mainly for checking – not considered an algebraic solution)
solve rational equations by changing to quadratic form
use the discriminant to find the number and type of solutions
A quadratic equation in x is an equation that can be written in the general form
2
ax bxc ,
where a, b, and c are real numbers, with a 0. A quadratic equation in x is also called a
second-degree polynomial equation.
The Zero-Product Principle: If the product of two algebraic expressions is zero, then at least one of
the factors equal to zero.
If AB = 0, then A = 0 or B = 0.
Solving Quadratic Equations by Factoring:
Steps:
Example 1: 9 x 12 x
2
- Rearrange equation so
that one side is 0.
(Look for GCF.)
- Factor. (Use sum/product
idea when a = 1. If
a 1 , use grouping*.)
- Set each factor equal to 0.
- Solve each equation.
- Check in original
equation or by graphing
(observe x-intercepts).
*Go to “Class Pages” on math home page (www.math.msu.edu) for steps using “Grouping Method”.
Example 2: 3 10
2 x x Example 3: ( x 1)( x 4) 10 (Can the Zero-
Product Principle be used in the first
step of the solution? Explain.)
Example 4: 4 13 3
2 x x (Use “guess & check” or “grouping method”.)
Example 5:
x 2
x
Example 6:
x 1 x 4 4
Example 11: Solve and simplify:
2 3 x 5 x 1
Example 12: Solve and simplify:
2 2 x 3 x 21
In the quadratic formula,
a
b b ac
x
2
, the value of^ b^4 ac
2 is called the discriminant.
Beware: The discriminant is NOT b 4 ac
2 !!!
2
b ac 4 0
2
b ac 4 0
2
b ac
Example 13: Refer to Examples 10 – 12 to complete the table below.
Example Discriminant Number and Type of Solutions
Example 14 (optional): Solve and simplify:
2 4 x 16 x 13 (Be sure to simplify radicals!)
Example 15: Applications of quadratic equations can be found in the medical field as well as many other
sciences. For example, consider a piece of artery or vein which is approximately the shape of a cylinder.
The velocity v at which blood travels through the arteries or veins is a function of the distance r of the
blood from the axis of symmetry of the cylinder. Specifically, for a wide arterial capillary the following
formula might apply:
4 2 v 1.185 (185 10 ) r , where r is measured in cm and v is measured in cm/sec.
(a) Sketch a picture of the cylinder including the axis of symmetry.
(b) Find the velocity of the blood traveling on the axis of symmetry of the capillary.
(c) Find the velocity of blood traveling
4 6 10
cm from the axis of symmetry.
(d) According to this model, where in the capillary is the velocity of the blood 0?
(e) What are the allowable values (domain) for r?
(f) Sketch a graph of this equation and interpret. Be sure to label your axes.
1.4 Application of Quadratic Equations
In this section you will learn to:
use quadratic equations to solve word problems
Example 1: The length of a rectangle exceeds its width
by 3 feet. If its area is 54 square feet, find its dimensions.
Example 2: The MSU football stadium currently has the
one of the largest HD video screens of any college stadium.
The rectangular screen’s length is 72 feet more than its height.
If the video screen has an area of 5760 square feet, find
the dimensions of the screen. (MSU math fact: The area
of the video screen is about 600 ft
2 larger than Breslin’s
basketball floor.)
Example 3: Find the dimensions of a rectangle whose area
is 180 cm
2 and whose perimeter is 54 cm.
Recall the Steps/Tips for Solving Word Problems used in Section 1.2:
- Read the problem carefully. Underline key words and phrases. Let x (or any variable)
represent one of the unknown quantities. Draw a picture or diagram if possible.
- If necessary write expressions for other unknowns in terms of x.
- Write a verbal model of the problem and then replace words with numbers, variables
and/or symbols.
- Solve the equation. Answer the question in the problem. Label answers!
- Check your answer(s) in the original word problem (not in your equation).
Example 4: When the sum of 8 and twice a positive
number is subtracted from the square of the number,
the result is 0. Find the number.
Example 5: In a round-robin tournament, each team is
paired with every team once. The formula below models
the number of games, N, that must be played in a tournament
with x teams. If 55 games were played in a round-robin
tournament, how many teams were entered?
2 x x
N
Example 6: Find at least two quadratic equations
whose solution set is
Example 7: A piece of sheet metal measuring 12 inches
by 18 inches is to have four equal squares cut from its
corners. If the edges are then to be folded up to make a
box with a floor area of 91 square inches, find the depth
of the box.
