Download Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 1 Name

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NOTES 1: EQUATIONS AND INEQUALITIES
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LESSON 1.1 – BASIC EQUATIONS
To solve rational
equations…
Multiply every term by the LCD of the whole equation. Only get rid of the denominator
when you are solving an equation.
Practice Problems: Solve the Following Equations.
1.
x3 2 4
 
x
x 3
2.
2
4

x3 x2
3.
2
1
3

 2
x  4 x  4 x  16
4.
2 x  1 12  3 x 2 x  1


x  4 x 2  16 x  4
2x 1 6  2x 2x 1


3x  2 9 x 2  4 3x  2
6.
5.
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 1
1
3
1
2w
 60
To solve equations where
one variable (or term) is
raised to a power…
Get the term with the exponent alone and then take a root of both sides or raise
both sides to the reciprocal exponent.
Practice Problems: Solve each equation. Find all the real solutions.
7.
3( x  2)2  27
8.
x 4  16  0
9.
3( x  3)3  375
10.
2 x 3  64  0
11.
( x  2)4  81  0
12.
5 x 3  2  43
5
2
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 2
LESSON 1.2 – MODELING WITH EQUATIONS
Steps for
Creating a
Model for a
Word Problem
1. Identify the variable
2. Express all unknown quantities in terms of the variable
3. Set up the model
4. Solve the equation and check your answer
Practice Problems: Set up an equation (or two) and then solve each problem.
1.
Jim scored 22 points in a basketball game. That 2.
The perimeter of a rectangle is 36 cm. The
was six points more than Frank scored. How many
length is 4 cm greater than the width. Find the width &
points did Frank score?
length.
3.
Marissa & Lisa earned a total of $65
babysitting during the month of November. Marissa
earned $5 more than half of what Lisa earned. How
much did they each earn?
4.
Sharon is 21 years older than Laura. In six
years, Sharon will be twice as old as Laura. How old
are they now?
5.
Four pencils & two pens cost $0.74. Six
pencils & five pens cost $1.53. Find the cost of one
pencil and the cost of one pen.
6.
A jar of quarters and nickels contains $1.25.
There are 13 coins in all. How many of each are there?
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 3
7.
San Diego Truck Rentals rents a truck at a
daily rate of $57.99 plus 48 cents per mile. Los
Angeles Truck Rentals rents the same truck for $58.95
plus 46 cents per mile. For what mileage is the cost the
same?
8.
A train leaves San Diego traveling east 80
km/hr. An hour later, another train leaves San Diego
on a parallel track at 120 km/hr. How far from San
Diego will the trains meet?
9.
A boat took 3 hours to make a downstream trip
with a current of 6 km/hr. The return trip, against the
same current, took 5 hours. Find the speed of the boat
in still water.
10.
Two cars leave town at the same time going in
opposite directions. One of them travels 60 mi/hr and
the other 30 mi/hr. In how many hours will they be
150 miles apart?
11.
One solution is 80% acid and another is 30%
acid. How much of each solution is needed to make a
200L solution that is 62% acid?
12.
A grocer wishes to mix some nuts worth 90
cents per pound with some nuts worth $1.60 per pound
to make 175 pounds of a mixture that is worth $1.30
per pound. How much of each should she use?
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 4
13.
Lydia needs an antifreeze solution that is 50%
alcohol. She has antifreeze that is 40% alcohol and
antifreeze that is 60% alcohol. How much of each
solution does she need to get 10 L of the antifreeze
that is 50% alcohol?
14.
Kristen can type a 50-page paper in 8 hours.
Last month, Kristen and Courtney, together types a
50-page paper in 6 hours. How long would it take
Courtney to type a 50 page paper on her own?
15.
It takes Al 3 hours to paint a certain area of the
house. It takes Sarah 5 hours to do the same job. How
long would it take them, working together, to do the
painting job?
16.
One machine in a print shop can produced a
certain number of pages twice as fast as another
machine. Operating together, these machines can
produce this number of pages in 8 minutes. How long
would it take each machine, working alone, to produce
this number of pages?
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 5
LESSON 1.3 – QUADRATIC EQUATIONS
Quadratic
Equation
Standard Form: y  ax 2  bx  c
Zero Product
Property
ab  0 iff a  0 or b  0
Vertex Form: y  a  x  h   k
2
Practice Problems: Solve each equation by factoring.
1.
x 2  x  30  0
2.
2 x 2  4 x  30  0
3.
6 x 2  33 x  15
4.
8 x 2  24 x  18
5.
12 x 2  3  0
6.
2 x 2  50  0
Completing the
Square
2
b
b 
x 2  bx      x  
2
2 
2
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 6
Example:
Steps for
Completing the
Square
1.
1.
Put the constants on the same side
of the equation as y.
2.
Factor out “a”.
3.
Take half of b and square the
value. Add that value to both sides
of the equation. Don’t forget that if
you factored “a” out then you need
2
b
to multiply   by a.
2
4.
Write the side with x as a perfect
square by factoring.
5.
Simplify the side with y.
Practice Problems: Solve each equation by completing the square.
7.
x2  6 x  5  0
8.
x2  2 x  2  0
9.
2 x2  4 x  7  0
10.
3x 2  9 x  4  5
11.
4 x2  6 x  1  0
12.
x2 
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 7
3
1
x
4
8
Quadratic Formula
If ax 2  bx  c  0 , then x 
b  b 2  4ac
2a
Practice Problems: Solve each equation by the quadratic formula.
13.
x 2  3x  8  0
14.
2 x2  3x  6  0
15.
3x 2  5 x  7  0
16.
9 x 2  30 x  25  0
Using the
Discriminant to
Determine the
Number and Type
of Solutions:

If b 2  4ac is negative, the equation has 2 imaginary solutions (no x-intercepts)

If b 2  4ac is zero, the equation has 1 real, rational solution (one x-intercept)

If b 2  4ac is positive and is a perfect square, the equation has 2 real, rational
solutions (2 x-intercepts at rational numbers)

If b 2  4ac is positive but is not a perfect square, the equation has 2 real,
irrational solutions (2 x-intercepts at irrational numbers)
Practice Problems: Use the discriminant to determine the number of real solutions of the equation. Do not
solve.
17.
x 2  3x  8  0
18.
9 x 2  30 x  25  0
19.
3x 2  5 x  7  0
20.
5 x 2  30 x  25  0
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 8
Practice Problems: Quadratic Modeling: Set up an equation for each problem & solve.
21.
An object shot straight upward at an initial
speed of v0 ft/s will reach a height of h feet after t
seconds, where h and t are related by the formula
22.
A rectangular building lot is 8 feet longer than
it is wide and has an area of 2900 ft 2 . Find the
dimensions of the lot.
h  16t 2  v0t . Suppose that a bullet is shot straight
upward with an initial speed of 800 ft/s.
a.
When does the bullet fall back to ground level?
b.
When does it reach a height of 6400 ft?
23.
Find two numbers whose sum is 40 and whose
product is 364.
24.
A rectangular swimming pool is 8 feet deep
everywhere and twice as long as it is wide. If the pool
holds 8464 ft 3 of water, what are the dimensions?
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 9
LESSON 1.4 – COMPLEX NUMBERS
Review Problems: Solve
1. x 2  1  0
2. x 2  8  0
i 2  1
i  1
Principal square root of a negative
number
Definition of a
complex number
Standard form:
i4  1
i 3   1  i
a  i a
a  bi
Examples:
Practice Problems: Rewrite then plot the following complex numbers.
1. 4  9 
y
2.
27 
x
3. 6i  i 2 
4. i 40 
5. i 57 
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 10
Adding Complex Numbers
Subtracting Complex
Numbers
Multiplying Complex
Numbers
Dividing Complex
Numbers
Add the real parts and the imaginary parts
Subtract the real parts and the imaginary parts
Multiply complex numbers like binomials, using i 2  1
Multiply the numerator and the denominator by the complex conjugate of the
denominator
Practice Problems: Perform the operations on the complex members.
6. 3  (2  3i )  (5  i )
7. (3  2i )  (4  i )  (7  i )
8. (2  i )(4  3i )
9. (3  2i ) 2
10.
1
1 i
11.
2  3i
4  2i
Practice Problems: Write the complex number in standard form and simplify.
12.
3 12
13.
48  27
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 11

14. 1  3

2
LESSON 1.5 – OTHER TYPES OF EQUATIONS
Practice Problems: Find all the real solutions of each equation.
1.
x5  27 x 2
2.
x 4  64 x  0
3.
x 4  x3  6 x 2  0
4.
x3  x 2  x  1  x 2  1
5.
x4  5x2  4  0
6.
x6  26 x3  27  0
7.
x 2  3x 4  4  0
8.
4( x  1) 2  5( x  1) 2  ( x  1) 2  0
1
1
1
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 12
3
5
9.
11.
x5
5
28


x  2 x  2 x2  4
x 1  2  4
2
x  5x
4
3
x
x
10.
12.
y  2 1  5
2 4x  8  4  8
13.
3  2x 3  5
14.
15.
x  4  x2  8
16.
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 13
2x  5  x  3  1
LESSON 1.6 – INEQUALITIES
Practice Problems: Solve the linear inequality. Express the solution using interval notation & graph the
solution set.
1.
5  8 x  19  10 x
2.
4  3 x  2  8
3.
5  2 x  7  13
4.
x  5  8 or x  5  8
5.
4  2 x  2 or 4  2 x  6
6.
1 4  3x 1
 

2
5
4
Quadratic
Inequalities
When solving non-linear inequalities, you need to solve the problem as if it were an equation,
set up intervals using the solutions, & test a value in each interval to see if the solution falls in
that interval.
Practice Problems: Solve the linear inequality. Express the solution using interval notation & graph the
solution set.
7.
Interval
x 2  3 x  18  0
Test Value
Polynomial Value
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 14
True/False?
3x 2  17 x  10  0
8.
Interval
Polynomial Value
True/False?
Test Value
Polynomial Value
True/False?
Test Value
Polynomial Value
True/False?
Test Value
Polynomial Value
True/False?
36  x 2  0
9.
Interval
x
 3x
x 1
10.
Interval
11.
Test Value
x
5

4
2 x 1
Interval
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 15
LESSON 1.7 – ABSOLUTE VALUE EQUATIONS AND INEQUALITIES
Absolute Value
Equations
When solving absolute value equations, you must set up 2 different equations since the
absolute value represents the distance that a number is from zero.
Practice Problems: Solve each equation.
1.
3 g  14  7
2.
6  3k  21
3.
3
1
x2  4
5
2
4.
x  1  3x  2
Absolute Value
Inequalities
To solve absolute value inequalities, you must set up 2 different inequalities and figure out
if the problem is an “and” or an “or” problem.
If |expression| > c then the problem is an “OR” problem.
If |expression| < c then the problem is an “AND” problem.
Practice Problems: Solve each inequality and graph your solutions on the number line.
5.
n  11  1
6.
19  5 x  7
7.
 3w  15  30
8.
0  x3  7
Mrs. Nguyen – Honors Algebra II – Chapter 1 Notes – Page 16