Download Chapter 1 Equations and Inequalities

Chapter 1 Equations and Inequalities
1.1 Linear Equations
Definition 1 An equation in one variable is a statement in which two expressions, at least
one containing the variable, are equal.
Example 1 Solve the equation 3x  5  4.
Definition 2 A linear equation in one variable is an equation that can be written in the
form
ax  b  0 ,
where a and b are real numbers and a ≠ 0.
Remark A linear equation is also called a first-degree equation.
Solving Linear Equations in One Variable
(1) Clear fractions: Multiply on both sides by the LCD to clear the equations of
fractions if they occur.
(2) Remove parentheses: Use the distributive property to remove parentheses if they
occur.
(3) Simplify: Simplify each side of the equation by combining like terms.
(4) Isolate the variable: Get all variable terms on one side and all numbers on the
other side by using the addition property of equality.
(5) Solve: Get the variable alone by using the multiplication property of equality.
Example 2 Solve the equation 3(2x – 4) = 7 – (x + 5).
Example 3 Solve the equation
1
1
( x  5)  4  (2 x  1).
2
3
1
Example 4 Solve the equation (2 y  1)( y  1)  ( y  5)( 2 y  5).
Example 5 Solve the equation
3
1
7


.
x  2 x  1 ( x  1)( x  2)
Example 6 Solve the equation
3x
3
2
.
x 1
x 1
2
Solving for a Specified Variable
All other letters are considered as numbers.
Example 7 Solve PV = nRT for T.
Example 8 Solve C 
5
( F  32) for F.
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Steps for Setting Up Applied Problems
(1) Read the problem carefully and identify what you are looking for.
(2) Assign a letter (variable) to represent what you are looking for, and, if necessary,
express any remaining unknown quantities in terms of this variable.
(3) Make a list of all the known facts, and translate them into mathematical
expressions. Set up the equation.
(4) Solve the equation for the variable, and then answer the question, usually using a
complete sentence.
(5) Check the answer with the facts in the problem.
Example 9 A total of $18,000 is invested, some in stocks and some in bonds, if the
amount invested in bonds is half that invested in stocks, how much is invested in each
category?
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Example 10 Shannon grossed $435 one week by working 52 hours. Her employer pays
time-and-a-half for all hours worked in excess of 40 hours. With this information, can
you determine Shannon’s regular hourly wage?
1.2Quadratic Equations
Definition 1 A quadratic equation is an equation which can be written in the form
ax 2  bx  c  0 ,
where a, b and c are real numbers and a ≠ 0.
Remark A quadratic equation written in the form of ax 2  bx  c  0 is said to be in
standard form.
Remark A quadratic equation is also called a second-degree equation.
Solving Quadratic Equations
1) Solving a Quadratic Equation by Factoring
Zero-product Property If ab  0 , then a=0 or b=0.
Example 1 Solve x 2  6 x  0.
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Example 2 Solve 2 x 2  x  3.
Remark When the left side factors into two linear equations with the same solution, the
quadratic equation is said to have a repeated solution. We also call this solution a root of
multiplicity 2, or a double root.
Example 3 Solve x 2  6 x  9  0.
2) The Square Root Method
If x 2  p and p  0 , then x 
Example 4 Solve (a) x 2  5
p or x   p .
(b) ( x  2) 2  16
3) Complete square
Procedure for completing a square
Start
Add
m
x 2  mx
( )2
2
Result
m
m
x 2  mx  ( ) 2  ( x  ) 2
2
2
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Example 5 Determine the number that must be added to each expression to complete the
square. Then factor the resulted expression.
(1) x 2  8 x
(2) x 2  20 x
Example 6 Solve by completing the square.
(1) x 2  5 x  4  0
(2) 2 x 2  8 x  5  0
4) The Quadratic Formula
For ax 2  bx  c  0 , a  0 , we have the quadratic formula
b 2  4ac
(1) If
(2) If
(3) If
 b  b 2  4ac
.
x
2a
is called the discriminant of the quadratic equation.
b 2  4ac  0 , there are two unequal real solutions.
b 2  4ac  0 , there is a repeated solution, a root of multiplicity 2.
b 2  4ac  0 , there is no solution.
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Example 7 Solve 3x 2  5 x  1  0
Example 8 Solve
25 2
x  30 x  18  0.
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Example 9 Solve 3x 2  2  4 x
7
Example 10 Solve 9 
3 2

 0, x  0
x x2
1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations
Definition 1 When the variable in an equation occurs in a radical (square root, cube root,
and so on), the equation is called a radical equation.
To solve a radical equation,
(1) Isolate the most complicated radical on one side of the equation.
(2) Eliminate it by raising each side to a power equal to the index of the radical.
(3) We need to check all answers when working with radical equations.
Example 1 Solve
3
2 x  4  2  0.
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Example 2 Solve
x  1  x  7.
Example 3 Solve
2 x  3  x  2  2.
Definition 2 If an appropriate substitution u transforms an equation in the form
au 2  bu  c  0, u ≠ 0
then the original equation is called an equation quadratic in form.
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Example 4 Solve ( x  2) 2  11( x  2)  12  0.
Example 5 Solve x 6  9 x 3  8  0.
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Example 6 Solve
3  2 x  x.
Factorable Equations
Example 7 Solve x 4  4 x 2 .
Example 8 Solve x 3  x 2  4 x  4  0.
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1.5 Solving Inequalities
Intervals
INTERVAL
Open interval (a, b)
Closed interval [a, b]
Half-open interval [a, b)
Half-open interval (a, b]
Interval [a, )
Interval (a, )
Interval (, a]
Interval ( , a )
Interval (, )
INEQUALITY
GRAPH
Example 1 Write each inequality using interval notation.
(a) 1  x  3 , (b)  4  x  0 , (c) x  5 , (d) x  1
Example 2 Write each interval as an inequality involving x
(a) [1,4), (b) (2,  ), (c) [2,3], (d) (- ,3 )
Properties of Inequalities
 Nonnegative Property: a 2  0 .
 Addition Property for Inequalities:
If a < b, then a + c < b + c.
If a > b, then a + c > b + c.
 Multiplication Properties for Inequalities:
If a < b and if c > 0, then ac < bc.
If a > b and if c > 0, then ac > bc.
If a < b and if c < 0, then ac > bc.
If a > b and if c < 0, then ac < bc.
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
Reciprocal Properties:
1
0.
a
1
If a  0 , then  0 .
a
If a  0 , then
Example 3 (a) If x  5 , then
(b) If x  2 , then
Example 4 (a) If 2x  6 , then
(b) If
x
 12 , then
3
(c) If  4x  8 , then
Solving Inequalities
Example 5 Solve the inequality 3  2x  5 .
Example 6 Solve the inequality 4x  7  2x  3 .
Solving Combined Inequalities
(1) Keep the variable in the middle.
(2) Work with all three expressions at the same time.
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Example 7 Solve the inequality  5  3x  2  1 .
Example 8 Solve the inequality  1 
3  5x
9.
2
Example 9 Solve the inequality (4 x  1) 1  0 .
Example 10 If  1  x  4 , find a and b so that a  2x  1  b .
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1.6 Equations and Inequalities Involving Absolute Value
Equations Involving Absolute Value
If a is a positive real number and if u is an algebraic expression, then
|u| = a is equivalent to u = a, or u = -a.
Example 1 Solve the equation x  4  13 .
Example 2 Solve the equation 2 x  3  2  7.
Example 3 Solve the equation x 2  x  1  1 .
Inequalities Involving Absolute Value (I)
If a is a positive real number and if u is an algebraic expression, then
|u| < a is equivalent to -a <u < a
|u|  a is equivalent to -a  u  a
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Example 4 Solve the inequality 2 x  4  3 .
Example 5 Solve the inequality 1  4 x  1  6 .
Inequalities Involving Absolute Value (II)
If a is a positive real number and if u is an algebraic expression, then
|u| > a is equivalent to u < -a or u > a
|u| ≥ a is equivalent to u  -a or u ≥ a
Example 6 Solve the inequality 2 x  5  3 .
Example 7 Solve the inequality x  3  1  1 .
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1.7 Applications: Interest, Mixture, Uniform Motion, Constant Rate Jobs
1) Solve Interest Problems
Simple Interest Formula If a principal of P dollars is borrowed for a period of t years at a per
annum interest rate r, expressed as a decimal, the interest I charged is
I = Prt.
Example 1 Candy has $70,000 to invest and requires an overall rate of return of 9%. She can
invest in a safe, government-insured certificate of deposit, but it only pays 8%. To obtain 9%, she
agrees to invest some of her money in noninsured corporate bonds paying 12%. How much
should be placed in each investment to achieve her goal?
2) Solve Mixture Problems
Example 2 The manager of a Starbucks store decides to experiment with a new blend of coffee.
She will mix some B grade coffee that sells for $5 per pound with some A grade coffee that sells
for $10 per pound to get 100 pounds of the new blend. The selling price of the new blend is to be
$7 per pound, and there is no difference in revenue from selling the new blend versus selling the
other types. How many pounds of the B grade and A grade coffees are required?
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3) Solve Uniform Motion Problems
Uniform Motion Formula If an object moves at an average velocity v, the distance s covered in
time t is given by
S = vt.
That is, Distance = Velocity ∙ Time
Example 3 Tanya, who is a long-distance runner, runs at an average velocity of 8 miles per hour
(mi/hr). Two hours after Tanya leaves your house, you leave in your Honda and follow the same
route. If your average velocity is 40mi/hr, how long will it be before you catch up to Tanya? How
far will each of you be from your home?
4) Solve Constant Rate Job Problems
If a job can be done in t units of time,
1
of the job is done in 1 unit of time.
t
Example 4
One computer can do a job twice as fast as another. Working together, both computers can do the
job in 2 hours. How long would it take each computer, working alone, to do the job?
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