Download Review for Algebra Final 2009

Review for Algebra Final 2013
4-1


Use rate of change to solve problems
Find the slope of a line
The table shows how the distance traveled changes with the number of hours driven.
Use the table to find the rate of change. Explain the meaning of the rate of change.
Time Driving (x)
Distance Traveled (y)
2
4
6
76
152
228
Find the slope of the line passing through:
1. (-3, -4) and (-2, -8)
-4
2. (-3, 4) and (4, 4)
0
38 miles per hour
Determine the value of n so the line that passes through each pair of points has the
given slope:
1. (6, 8), (n, -2), m = 1
2. (n, 4), (7, 1), m =
3
n = 11
4
3. (n, 3), (7, n), m =
1
5
4-2


n = -4
n=2
Write and graph direct variation equations
Solve problems involving direct variation
Graph the following equations:
(see next page)
1
1. y = 2x
2. y =
2
x
3
2
Write a direct variation equation:
1. Suppose y varies directly as x, and y = 9 when x = -3. Write a direct variation
equation that relates x and y. Then use the equation to find x when y = 15.
y = -3x
x = -5
2. Suppose y varies directly as x, and y = 15 when x = 5. Write a direct variation
equation that relates x and y. Then use the equation to find x when y = -45.
y = 3x
4-3


x = -15
Write and graph linear equations in slope-intercept form
Model real-world data with an equation in slope-intercept form
Write an equation of these lines:
1. slope 8; y-intercept -3
y = 8x - 3
2. slope -1; y-intercept 4
y = -x + 4
Graph these equations:
1. y = 2x – 1
2. y = -x + 4
3
Use this information for the next 3 questions: The population of the United States is
projected to be 300 million by the year 2010. Between 2010 and 2050, the population
is expected to increase by about 2.5 million per year.
1. Write an equation to find the population P in any year x between 2010 and 2050.
P = 2.5(x - 2010) + 300
2. Graph the equation.
3. Find the population in 2050.
400 million
y-axis represents population (by hundred million)
x-axis represents year (0=2010, 1=2020, etc.)
Write the equation of the line shown in each graph.
(see next page)
4
4-4


y=x+1
y=
5
1
x2
2
2
Write an equation of a line given the slope and one point
Write an equation of a line given two points
Write an equation of the line that passes through the following points with the given
slope:
3
4
1
2. (4, -5), m =
2
1. (8, 2), m =
3
x8
4
1
x3
y=
2
y=
Write an equation of the line that passes through the following points:
1. (0, 2), (1, 7)
y = 5x + 2
2. (-2, -1), (2, 11)
y = 3x + 5
4-5



Write equations in point-slope form
Write equations in slope-intercept form
Write equations in standard form
Write the point-slope form of an equation for a line:
1. passing through (6, 1) with a slope of
5
2
y–1=
5
(x – 6)
2
2. passing through (-7, 2) with a slope of 6
y – 2 = 6(x + 7)
3. a horizontal line that passes through (4, -1) y + 1 = 0
5
Write each equation in standard form:
1. y + 2 = -3(x – 1)
3x + y = 1
2. y – 4 =
5
( x  3)
3
5x – 3y = -27
Write each equation in slope-intercept form:
1. y + 4 = 4(x – 2)
y = 4x - 12
2. y – 8 =
4-7

1
( x  8)
4
y=
1
x+6
4
Write equations for parallel and perpendicular lines
Write the slope-intercept form for an equation of the line that passes through the given
point and is parallel to the given line:
1. (-2, 2); y = 4x -2
y = 4x + 10
2. (-1, 6); 3x + y = 12 y = -3x + 3
3. (-3, 4); y = 6
y=4
4. (-3, 4); x = 7
x = -3 (this is not slope-intercept form)
Write the slope-intercept form for an equation of the line that passes through the given
point and is perpendicular to the given line:
1. (4, 2); y =
1
x 1
2
y = -2x + 10
1
x-4
3
2. (6, -6); 3x – y = 6
y=
3. (5, -2); y = 6
4. (5, -2); x = 7
x = 5 (this is not slope-intercept form)
y = -2
5-1


Determine whether a system of linear equations has no, one, or infinitely many
solutions
Solve systems of equations by graphing
Use the graph to determine whether the system has no solution (inconsistent), one
solution (consistent and independent), or infinitely many solutions (consistent and
dependent).
1. y = -x + 2
y=x+1
2. y = -x + 2
3x + 3y = -3
1 solution (consistent and independent)
no solution (inconsistent – lines are parallel)
3. 3x + 3y = -3 infinitely many solutions (consistent and dependent – graphs of
y = -x – 1
the lines are the same)
6
Graph each system of equations. Then determine whether the system has no solution,
one solution, or infinitely many solutions. If the system has one solution, name it.
1. y = -2
-2x – y = -2
(2, -2)
2. y = -4x + 4
3 2
 ,2 
7 7
3x – y = -1
7
3. 3x + 2y = 6
2y = -3x – 4
no solution (parallel)
5-2
1. Solve systems of equations by using substitution
Solve each system using substitution:
1. x = -4y
3x + 2y = 20
(8, -2)
2. x + 5y = 4
3x + 15y = -1
no solution
3. x + 4y = 8
2x – 5y = 29
(12, -1)
4. x – 5y = 10
-10y = -2x + 20
infinitely many solutions
8
5-3

Solve systems of equations using elimination
Solve each system using elimination:
1. x – 2y = 6
x+y=3
(4, -1)
2. 4x + y = 23
3x – y = 12
(5, 3)
3. Four times one number added to another number is 36. Three times the first
number minus the other number is 20. Find the numbers.
4x + y = 36
3x – y = 20
8 and 4
5-4

Solve systems of equations using elimination with multiplication
Use elimination to solve each system of equations:
1. 2x + 3y = 6
x + 2y = 5
(-3, 4)
2. 3a – b = 2
a + 2b = 3
(1, 1)
3. 2x + y = 3.5
-x + 2y = 2.5
(0.9, 1.7)
4. 4x + 2y = -5
-2x – 4y = 1
(-3/2, ½)
9
6-1


Solve linear inequalities by using addition
Solve linear inequalities by using subtraction
Solve each inequality. Then graph it on the number line:
1. n – 8 < -13
n < -5 open circle on -5, shade to left
2. -6 ≥ n – 3
n < -3 closed circle on -3, shade to left
3. 3n -1 < 4n
n > -1 open circle on -1, shade to right
6-2


Solve linear inequalities by using multiplication
Solve linear inequalities by using division
Solve each inequality:
1.
n
 22
50
n ≤ -1100
2.
3m  3

5
20
m ≤-1/4
3. 18 < -3b
b < -6
4. -40 > 10h
h < -4
10
6-3


Solve linear inequalities involving more than one operation
Solve linear inequalities involving the distributive property
Solve each inequality:
1. -12 – d > -12 + 4d
d<0
2. k – 17 ≤ - (17 – k)
all numbers
3. 2(y – 2) > -4 + 2y
no solution
4. 8 – 2(b + 1) < 12 – 3b
b<6
6-4


Solve compound inequalities containing the word and and graph their solution
sets
Solve compound inequalities containing the word or and graph their solution sets
Solve each inequality. Then graph the solution set:
1. 4 < w + 3 < 5
1 < w < 2 open circles on 1 & 2 shaded between
2. -3p + 1 < -11 or p < 2
p > 4 or p < 2 open circles on 4 & 2 shaded outward
3. 2y + 2 < 12 or y – 3 > 2y
y < 5 or y < -3 open circle on 7 shaded to left
6-5


Solve absolute value equations
Graph absolute value functions
Solve each open sentence. Then graph the solution set:
11
1. |x – 4| = 5
2. |2x – 1| = 11
x = 9, -1 points on 9, -1
x = 6, -5 points on 6, -5
3. Grant’s highest and lowest test scores are 7 percentage points from 89%. Write
and solve an open sentence to determine his scores. |g - 89| = 7; 82% & 96%
4. The normal human body temperature varies 1.5º from the average 98.6ºF.
Write and solve an open sentence to determine the variation in temperature.
|t – 98.6| = 1.5; 100.1° & 97.1°
6-6


Solve absolute value inequalities
Apply absolute value inequalities in real-world situations
Solve each open sentence. Then graph the solution set:
1. |c – 2| > 6
c > 8 or c < -4 open circles on 8 & -4 shaded outward
2. |2d – 1| < 4
d < 2.5 and d > -1.5 open circles on 2.5 & -1.5 shaded between
3. |x – 3| < 0
no solution
4. |2x + 1| > -2 all numbers
12
6-7


Graph inequalities on the coordinate plane
Solve real-world problems involving linear inequalities
Graph:
3x < y shade above dotted line
3x + 6y > 12 shade above dotted line
6-8

Solve systems of inequalities by graphing
Solve each system by graphing:
1. y < x + 1 and
2. y < 2x + 3 and
3x + 4y > 12 shade to the right of intersection of dotted lines in
quadrant I
y > -1 + 2x
shade between the dotted lines
13
7-1
2. Multiply monomials
3. Simplify expressions involving powers of monomials
Simplify:
1. (-7x2)(x4)
2. (-5xy)(4x2)(y4)
3. (3a3n4)(-3a3n)4
4. (2x3y2z2)3(x2z)4
-7x6
-20x3y5
243a15n8
8x17y6z10
7-2


Simplify expressions involving the quotient of monomials
Simplify expressions containing negative exponents
Simplify:
1.
 3r 6 s 3 

2. 
5
 2r s 
3.
 x7
7 y5
 2x 7
14 y 5
p 8
p3
4
81 4 8
rs
16
1
p 11
14
 4m 2 n 2
4. 
1
 8m



0
 2mn 
2 3
5.
4m 6 n 4
1
 m3
32n10
7-3


Find the degree of a polynomial
Arrange the terms of a polynomial in ascending or descending order
State whether each expression is a polynomial. If the expression is a polynomial,
identify it as a monomial, binomial, or trinomial:
1. 36
yes; monomial
2. 7x – x + 5
yes; binomial
3. 8xy2 + y2
yes; binomial
4.
2
+ 5y – 7
3y 2
no
Find the degree of each polynomial:
1.
2.
3.
4.
12n
1
45
0
2x2y5z
8
3
2
n p + 4np – 3n5
5
7-4


Add polynomials
Subtract polynomials
Find each sum:
1. (4a – 5) + (3a + 6)
7a + 1
2
2
2
2. (x + 2xy + y ) + (x – xy - 2y2)
2x2 + xy – y2
Find each difference:
1. (9x + 2) – (-3x2 – 5)
3x2 + 9x + 7
2
2
2
2. (4x + 6xy + 2y ) – (-x + 2xy – 5y2)
5x2 + 4xy + 7y2
7-5
15


Find the product of a monomial and a polynomial
Solve equations involving polynomials
Find each product:
1. 3x(5x2 – x + 4)
2. -3n2(-2n2 + 3n + 4)
15x3 – 3x2 + 12x
6n4 – 9n3 – 12n2
Simplify:
1. w(3w + 2) + 5w
2. 4b(-5b – 3) -2(b2 – 7b – 4)
3w2 + 7w
-22b2 + 2b + 8
Solve:
1. 3(a + 2) + 5 = 2a + 4
a = -7
2. 6(m – 2) + 14 = 3(m + 2) – 10
m = -2
7-6

Multiply two polynomials
Find each product:
1. (x – 4)(x + 1)
x2 – 3x - 4
2. (5t + 4)(2t – 6)
10t2 – 22t - 24
2
3. (2x – 1)(x – x + 2) 2x3 – 3x2 + 5x - 2
4. (y2 – 5y + 3)(2y2 + 7y – 4) 2y4 – 3y3 – 33y2 + 41y - 12
7-7


Find square of sums and squares of differences
Find the product of a sum and a difference
Find each product:
1. (x – 6)2
2. (x2 + 1)2
1

x  3
4

2
2

4.  x  2 
3

2
3. 
x2 – 12x + 36
x4 + 2x2 + 1
1 2 3
x  x9
16
2
4 2 8
x  x4
9
3
5. (2x – 1)(2x + 1)
6. (8 + 4x)(8 – 4x)
7. (2p – 5s)(2p + 5s)
4x2 - 1
64 – 16x2
4p2 – 25s2
16
4
 4

x  2 y  x  2 y 
3
 3

16 2
x  4y2
9
7. 
8-1


Find prime factorizations of monomials
Find the greatest common factors of monomials
Factor each monomial completely:
1. 600
23(3)(52)
2
2. 18m n
2(32)mmn
Find the GCF of each set of monomials:
1. 64, 80
16
2
2. 49x, 343x
49x
3. 12x2, 32x2yz, 60xy2
4x
8-2



Factor using the distributive property
Factor polynomials by grouping
Solve quadratic equations of the form ax2 + by = 0
Factor:
1. 24x + 48y
24(x + 2y)
2. 14c3 – 42c5 – 49c4
7c3(2 - 6c2 – 7c)
3. 4a2b + 28ab2 + 7ab ab(4a + 28b + 7)
4. 12ax + 3xz + 4ay + yz
(3x + y)(4a + z)
5. 12a2 + 3a – 8a – 2
(4a + 1)(3a – 2)
2
2
6. 4m + 4mn + 3mn + 3n
(4m + 3n)(m + n)
Solve each equation. Check your solutions:
1. 3m(m – 4) = 0
0, 4
2. (4m + 8)(m – 3) = 0 -2, 3
3. 4y2 = 28y
0, 7
2
4. 4m = 4m
0, 1
8-3


Factor trinomials of the form x2 + bx + c
Solve equations of the form x2 + bx + c = 0
Factor:
1. x2 – 4x – 21
(x – 7)(x + 3)
17
2. m2 + 9m + 20
3. a2 – 4ab + 4b2
4. x2 + 12x + 20
(m + 4)(m + 5)
(a – 2b)(a – 2b)
(x + 10)(x + 2)
Solve each equation. Check
1. y2 – 5y + 4 = 0
2. p2 = 9p – 14
3. a2 – 18a = -72
your solutions:
1, 4
2, 7
6, 12
8-4


Factor trinomials of the form ax2 + bx + c
Solve equations of the form ax2 + bx + c = 0
Factor:
1. 3m2 – 8m - 3
2. 18y2 + 9y - 5
3. 8x2 – 4x – 24
4. 4x2 + 26x – 48
(3m + 1)(m – 3)
(6y + 5)(3y – 1)
4(x – 2)(2x + 3)
2(x + 8)(2x – 3)
Solve each equation. Check
1. 3n2 – 2n – 5 = 0
2. 3x2 – 13x = 10
3. 2p2 = -21p – 40
4. 4a2 – 18a + 5 = 15
your solutions:
-1, 5/3
-2/3, 5
-8, -5/2
5, -1/2
8-5


Factor binomials that are differences of squares
Solve equations involving differences of squares
Factor:
1. m2 – 100
2. 16a2 – 9b2
3. 72p2 – 50
4. 2a3 – 98ab2
(m + 10)(m – 10)
(4a – 3b)(4a + 3b)
2(6p + 5)(6p – 5)
2a(a – 7b)(a + 7b)
Solve each equation. Check
1. 36n2 = 1
2. 25d2 – 100 = 0
3. 2m3 = 32m
your solutions:
1/6, -1/6
2, -2
0, 4, -4
8-6


Factor perfect square trinomials
Solve equations involving perfect squares
18
Determine whether each trinomial is a perfect square trinomial. If so, factor it.
1. x2 – 16x + 64
yes (x – 8)(x – 8)
2. p2 + 8p + 64
no
2
3. 36x - 12x + 1
(6x – 1)(6x – 1)
4. 16a2 – 40ab + 25b2 (4a – 5b)(4a – 5b)
Solve each equation. Check your solutions:
1. 16n2 + 16n + 4 = 0 -1/2
2. x2 + 10x + 25 = 0
-5
3. x2 + 30x + 150 = -75
-15
Problem Solving
 Distance problems
 Age problems
 Mixture problems
 Consecutive integer problems
Solve the following problems:
1. Two planes are traveling toward each other and are 950 miles apart. One plane
is flying at 170 mph. The other is flying at 210 mph. In how many hours will
the planes pass each other? 2.5 hours
2. A plane on a search mission flew east from an airport, turned, and flew west
back to the airport. The plane cruised at 200 km/h when flying east and 300
km/h when flying west. If the plane was in the air for 6 hours, how far from the
airport did it travel? 720 km
3. With the wind, a jet can fly 1500 miles in 2 hours 30 minutes. Against the wind,
it can fly only 1200 miles in the same time. Find the rate of the jet in still air and
the rate of the wind. 540 mph; 60 mph
4. When a plane flies into the wind, it can travel 3000 miles in 6 hours. When it
flies with the wind, it can travel the same distance in 5 hours. Find the rate of
the plane in still air and the rate of the wind.
550 mph; 50 mph
5. Omega is four years older than Sigma. In seven years, she will be twice as old
as Sigma was five years ago. Find their ages now. 21, 25
6. Mr. Munch is twice as old as his daughter. In two years, he will be four times as
old as she was fifteen years ago. How old are they now? 31, 62
7. GORP Corp. mixes raisins that cost $5 per pound with peanuts that cost $3.70
per pound. How many pounds of raisins should be combined with 10 pounds of
peanuts to make a mixture worth $4 per pound? 3 pounds
8. Going Nuts, Inc., sells cashews for $7 per pound and almonds for $4 per pound.
How many pounds of each should be combined to get 15 pounds of a mixture
worth $5 per pound? 10 lb. almonds and 5 lb. cashews
9. Find two consecutive even integers such that the sum of the larger and 3 times
the smaller is 234.
58, 60
10. Find three consecutive odd integers such that the sum of 7 times the smallest
and twice the largest is -91. -11, -9, -7
11. Find two consecutive even numbers whose product is 224. 14, 16
19
12. The product of two consecutive odd numbers is 143. Find their sum.
24, -24
Challenge Problems:
13. With the wind, a jet can fly 1500 miles in 2 hours 30 minutes. Against the wind,
it can fly only 1200 miles in the same time. Find the rate of the jet in still air and
the rate of the wind. 540 mph; 60 mph
14. When a plane flies into the wind, it can travel 3000 miles in 6 hours. When it
flies with the wind, it can travel the same distance in 5 hours. Find the rate of
the plane in still air and the rate of the wind.
550 mph; 50 mph
15. Omega is four years older than Sigma. In seven years, she will be twice as old
as Sigma was five years ago. Find their ages now. 21, 25
16. Mr. Munch is twice as old as his daughter. In two years, he will be four times as
old as she was fifteen years ago. How old are they now? 31, 62
20