A.3 Equations, Inequalities, and Problem Solving
Tải bản đầy đủ - 0trang
A10
Appendix A
Review of Elementary Algebra Topics
EXAMPLE 2
Solving a Linear Equation in Nonstandard Form
Solve 5x ϩ 4 ϭ 3x Ϫ 8.
Solution
5x ϩ 4 ϭ 3x Ϫ 8
5x Ϫ 3x ϩ 4 ϭ 3x Ϫ 3x Ϫ 8
2x ϩ 4 ϭ Ϫ8
2x ϩ 4 Ϫ 4 ϭ Ϫ8 Ϫ 4
Write original equation.
Subtract 3x from each side.
Combine like terms.
Subtract 4 from each side.
2x ϭ Ϫ12
Combine like terms.
2x Ϫ12
ϭ
2
2
Divide each side by 2.
x ϭ Ϫ6
Simplify.
The solution is x ϭ Ϫ6. Check this in the original equation.
Linear equations often contain parentheses or other symbols of grouping. In
most cases, it helps to remove symbols of grouping as a first step in solving an
equation. This is illustrated in Example 3.
EXAMPLE 3
Solving a Linear Equation Involving Parentheses
Solve 2͑x ϩ 4͒ ϭ 5͑x Ϫ 8͒.
Solution
2͑x ϩ 4͒ ϭ 5͑x Ϫ 8͒
2x ϩ 8 ϭ 5x Ϫ 40
2x Ϫ 5x ϩ 8 ϭ 5x Ϫ 5x Ϫ 40
Ϫ3x ϩ 8 ϭ Ϫ40
Study Tip
Recall that when finding the least
common multiple of a set of
numbers, you should first consider
all multiples of each number. Then,
you should choose the smallest of
the common multiples of the
numbers.
Ϫ3x ϩ 8 Ϫ 8 ϭ Ϫ40 Ϫ 8
Write original equation.
Distributive Property
Subtract 5x from each side.
Combine like terms.
Subtract 8 from each side.
Ϫ3x ϭ Ϫ48
Combine like terms.
Ϫ3x Ϫ48
ϭ
Ϫ3
Ϫ3
Divide each side by Ϫ3.
x ϭ 16
Simplify.
The solution is x ϭ 16. Check this in the original equation.
To solve an equation involving fractional expressions, find the least common
multiple (LCM) of the denominators and multiply each side by the LCM.
Section A.3
Equations, Inequalities, and Problem Solving
Solving a Linear Equation Involving Fractions
EXAMPLE 4
Solve
A11
x
3x
ϩ
ϭ 2.
3
4
Solution
12
12 и
3x ϩ 3x4 ϭ 12͑2͒
Multiply each side of original
equation by LCM 12.
x
3x
ϩ 12 и
ϭ 24
3
4
Distributive Property
4x ϩ 9x ϭ 24
Clear fractions.
13x ϭ 24
xϭ
Combine like terms.
24
13
Divide each side by 13.
The solution is x ϭ 24
13 . Check this in the original equation.
To solve an equation involving an absolute value, remember that the expression
inside the absolute value signs can be positive or negative. This results in two
separate equations, each of which must be solved.
Solving an Equation Involving Absolute Value
EXAMPLE 5
Խ
Խ
Solve 4x Ϫ 3 ϭ 13.
Solution
Խ4x Ϫ 3Խ ϭ 13
Write original equation.
4x Ϫ 3 ϭ Ϫ13 or 4x Ϫ 3 ϭ 13
4x ϭ Ϫ10
xϭϪ
5
2
4x ϭ 16
xϭ4
Equivalent equations
Add 3 to each side.
Divide each side by 4.
The solutions are x ϭ Ϫ 52 and x ϭ 4. Check these in the original equation.
Inequalities
The simplest type of inequality is a linear inequality in one variable. For instance,
2x ϩ 3 > 4 is a linear inequality in x. The procedures for solving linear
inequalities in one variable are much like those for solving linear equations, as
described on page A9. The exception is that when each side of an inequality is
multiplied or divided by a negative number, the direction of the inequality
symbol must be reversed.
A12
Appendix A
Review of Elementary Algebra Topics
Solving a Linear Inequality
EXAMPLE 6
Solve and graph the inequality Ϫ5x Ϫ 7 > 3x ϩ 9.
Solution
Ϫ5x Ϫ 7 > 3x ϩ 9
Write original inequality.
Ϫ8x Ϫ 7 > 9
Ϫ8x > 16
x < −2
−3
−2
−1
Add 7 to each side.
Divide each side by Ϫ8 and reverse the
direction of the inequality symbol.
x < Ϫ2
x
−4
Subtract 3x from each side.
The solution set in interval notation is ͑Ϫ ϱ, Ϫ2͒ and in set notation is
ͭx x < Ϫ2ͮ. The graph of the solution set is shown in Figure A.8.
0
Խ
Figure A.8
Two inequalities joined by the word and or the word or constitute a compound
inequality. Sometimes it is possible to write a compound inequality as a double
inequality. For instance, you can write Ϫ3 < 6x Ϫ 1 and 6x Ϫ 1 < 3 more
simply as Ϫ3 < 6x Ϫ 1 < 3. A compound inequality formed by the word and is
called conjunctive and may be rewritten as a double inequality. A compound
inequality joined by the word or is called disjunctive and cannot be rewritten as
a double inequality.
EXAMPLE 7
Solving a Conjunctive Inequality
Solve and graph the inequality 2x ϩ 3 Ն 4 and 3x Ϫ 8 < Ϫ2.
Solution
2x ϩ 3 Ն 4
1
2
1
2
≤ x<2
x
−1
0
1
2
3
Study Tip
2x Ն 1
3x < 6
1
2
x < 2
x Ն
x > 5
x>5
Figure A.10
2
3
4
5
Յ x < 2ͮ.
Solving a Disjunctive Inequality
Solution
6
or
Ϫ6x ϩ 1 Ն Ϫ5
Ϫ6x Ն Ϫ6
x Յ 1
x
1
1
2
Solve and graph the inequality x Ϫ 8 > Ϫ3 or Ϫ6x ϩ 1 Ն Ϫ5.
x Ϫ 8 > Ϫ3
0
Խ
The solution set in interval notation is ͓2, 2͒ and in set notation is ͭ x
The graph of the solution set is shown in Figure A.9.
EXAMPLE 8
Recall that the word or is
represented by the symbol ʜ,
which is read as union.
−1
3x Ϫ 8 < Ϫ2
1
Figure A.9
x≤1
and
The solution set in interval notation is ͑Ϫ ϱ, 1͔ ʜ ͑5, ϱ͒ and in set notation is
ͭx x > 5 or x Յ 1ͮ. The graph of the solution set is shown in Figure A.10.
Խ
Section A.3
Equations, Inequalities, and Problem Solving
A13
To solve an absolute value inequality, use the following rules.
Solving an Absolute Value Inequality
Let x be a variable or an algebraic expression and let a be a real number
such that a > 0.
1. The solutions of x < a are all values of x that lie between Ϫa and a.
x < a if and only if Ϫa < x < a
ԽԽ
ԽԽ
ԽԽ
2. The solutions of x > a are all values of x that are less than Ϫa or
greater than a.
x > a if and only if x < Ϫa or x > a
ԽԽ
These rules are also valid if < is replaced by Յ and > is replaced by Ն .
Solving Absolute Value Inequalities
EXAMPLE 9
Solve and graph each inequality.
a. 4x ϩ 3 > 9
b. 2x Ϫ 7 Յ 1
Խ
Խ
Խ
Խ
Solution
a. 4x ϩ 3 > 9
Խ
Խ
Write original inequality.
4x ϩ 3 < Ϫ9 or
4x < Ϫ12
x < Ϫ3
4x ϩ 3 > 9
4x > 6
x > 32
Equivalent inequalities
Subtract 3 from each side.
Divide each side by 4.
The solution set consists of all real numbers that are less than Ϫ3 or greater
than 32. The solution set in interval notation is ͑Ϫ ϱ, Ϫ3͒ ʜ ͑32, ϱ͒ and in set
notation is ͭ x x < Ϫ3 or x > 32ͮ. The graph is shown in Figure A.11.
Խ
x < −3
x > 32
3
2
x
−4 −3 −2 −1
0
1
2
3
Figure A.11
Խ
Խ
b. 2x Ϫ 7 Յ 1
Write original inequality.
Ϫ1 Յ 2x Ϫ 7 Յ 1
3≤ x≤4
x
1
2
Figure A.12
3
4
5
Equivalent double inequality
6 Յ 2x Յ 8
Add 7 to all three parts.
3 Յ x Յ 4
Divide all three parts by 2.
The solution set consists of all real numbers that are greater than or equal to 3
and less than or equal to 4. The solution set in interval notation is ͓3, 4͔ and in
set notation is ͭx 3 Յ x Յ 4ͮ. The graph is shown in Figure A.12.
Խ
A14
Appendix A
Review of Elementary Algebra Topics
Problem Solving
Algebra is used to solve word problems that relate to real-life situations. The
following guidelines summarize the problem-solving strategy that you should use
when solving word problems.
Guidelines for Solving Word Problems
1. Write a verbal model that describes the problem.
2. Assign labels to fixed quantities and variable quantities.
3. Rewrite the verbal model as an algebraic equation using the assigned
labels.
4. Solve the resulting algebraic equation.
5. Check to see that your solution satisfies the original problem as stated.
EXAMPLE 10
Finding the Percent of Monthly Expenses
Your family has an annual income of $77,520 and the following monthly expenses:
mortgage ($1500), car payment ($510), food ($400), utilities ($325), and credit
cards ($300). The total expenses for one year represent what percent of your
family’s annual income?
Solution
The total amount of your family’s monthly expenses is
1500 ϩ 510 ϩ 400 ϩ 325 ϩ 300 ϭ $3035.
The total monthly expenses for one year are
3035 и 12 ϭ $36,420.
Verbal
Model:
Expenses ϭ Percent
и
Income
Labels:
Expenses ϭ 36,420
Percent ϭ p
Income ϭ 77,520
Equation:
36,420 ϭ p и 77,520
Original equation
36,420
ϭp
77,520
Divide each side by 77,520.
0.470 Ϸ p
(dollars)
(in decimal form)
(dollars)
Use a calculator.
Your family’s total expenses for one year are approximately 0.470 or 47.0% of
your family’s annual income.
Section A.3
EXAMPLE 11
A15
Equations, Inequalities, and Problem Solving
Geometry: Similar Triangles
To determine the height of the Aon Center Building (in Chicago), you measure
the shadow cast by the building and find it to be 142 feet long, as shown in Figure
A.13. Then you measure the shadow cast by a four-foot post and find it to be
6 inches long. Estimate the building’s height.
x ft
Solution
To solve this problem, you use a property from geometry that states that the ratios
of corresponding sides of similar triangles are equal.
48 in.
Height of building
Verbal
Model:
Height of post
ϭ
Length of building’s shadow
Length of post’s shadow
6 in.
142 ft
Not drawn to scale
Height of building ϭ x
Length of building’s shadow ϭ 142
Height of post ϭ 4 feet ϭ 48 inches
Length of post’s shadow ϭ 6
Labels:
Figure A.13
x
48
ϭ
142
6
Proportion:
x и 6 ϭ 142 и 48
x ϭ 1136
(feet)
(feet)
(inches)
(inches)
Original proportion
Cross-multiply
Divide each side by 6.
So, you can estimate the Aon Center Building to be 1136 feet high.
EXAMPLE 12
Geometry: Dimensions of a Room
A rectangular kitchen is twice as long as it is wide, and its perimeter is 84 feet.
Find the dimensions of the kitchen.
w
l
Figure A.14
Solution
For this problem, it helps to sketch a diagram, as shown in Figure A.14.
Verbal
Model:
Labels:
2
и
Length ϩ 2 и Width ϭ Perimeter
Length ϭ l ϭ 2w
Width ϭ w
Perimeter ϭ 84
Equation: 2͑2w͒ ϩ 2w ϭ 84
Study Tip
For more review on equations and
inequalities, refer to Chapter 3.
6w ϭ 84
w ϭ 14
(feet)
(feet)
(feet)
Original equation
Combine like terms.
Divide each side by 6.
Because the length is twice the width, you have l ϭ 2w ϭ 2͑14͒ ϭ 28. So, the
dimensions of the room are 14 feet by 28 feet.
A16
Appendix A
Review of Elementary Algebra Topics
᭤ The Rectangular Coordinate System
᭤ Graphs of Equations
A.4 Graphs and Functions
᭤ Functions
᭤ Slope and Linear Equations
The Rectangular Coordinate System
᭤ Graphs of Linear Inequalities
You can represent ordered pairs of real numbers by points in a plane. This plane
is called a rectangular coordinate system. A rectangular coordinate system is
formed by two real lines, the x-axis (horizontal line) and the y-axis (vertical line),
intersecting at right angles. The point of intersection of the axes is called the
origin, and the axes divide the plane into four regions called quadrants.
Each point in the plane corresponds to an ordered pair ͑x, y͒ of real numbers
x and y, called the coordinates of the point. The x-coordinate tells how far to the
left or right the point is from the vertical axis, and the y-coordinate tells how far
up or down the point is from the horizontal axis, as shown in Figure A.15.
y
Quadrant II
4
Quadrant I
3
2
y-axis
y-coordinate
1
−4 −3 −2 −1
−1
x-axis −2
Quadrant III
(3, 2)
x-coordinate
−3
−4
x
1
2
3
4
Origin
Quadrant IV
Figure A.15
y
EXAMPLE 1
4
C
B
3
2
Determine the coordinates of each of the points shown in Figure A.16, and then
determine the quadrant in which each point is located.
D
1
− 4 − 3 −2 −1
−1
E
−2
−3
−4
Figure A.16
x
1
2
A
3
4
Finding Coordinates of Points
Solution
Point A lies two units to the right of the vertical axis and one unit below the
horizontal axis. So, point A must be given by ͑2, Ϫ1͒. The coordinates of the
other four points can be determined in a similar way. The results are as follows.
Point
A
B
C
D
E
Coordinates
͑2, Ϫ1͒
͑4, 3͒
͑Ϫ1, 3͒
͑0, 2͒
͑Ϫ3, Ϫ2͒
Quadrant
IV
I
II
None
III
Section A.4
A17
Graphs and Functions
Graphs of Equations
The solutions of an equation involving two variables can be represented by points
on a rectangular coordinate system. The graph of an equation is the set of all
points that are solutions of the equation.
The simplest way to sketch the graph of an equation is the point-plotting
method. With this method, you construct a table of values consisting of several
solution points of the equation, plot these points, and then connect the points with
a smooth curve or line.
Sketching the Graph of an Equation
EXAMPLE 2
Sketch the graph of y ϭ x 2 Ϫ 2.
Solution
Begin by choosing several x-values and then calculating the corresponding
y-values. For example, if you choose x ϭ Ϫ2, the corresponding y-value is
y ϭ x2 Ϫ 2
Original equation
y ϭ ͑Ϫ2͒ Ϫ 2
Substitute Ϫ2 for x.
y ϭ 4 Ϫ 2 ϭ 2.
Simplify.
2
Then, create a table using these values, as shown below.
x
y؍
x2
؊2
Ϫ2
Ϫ1
0
1
2
3
2
Ϫ1
Ϫ2
Ϫ1
2
7
͑Ϫ1, Ϫ1͒
͑0, Ϫ2͒
͑1, Ϫ1͒
͑2, 2͒
͑3, 7͒
͑Ϫ2, 2͒
Solution point
Next, plot the solution points, as shown in Figure A.17. Finally, connect the
points with a smooth curve, as shown in Figure A.18.
y
y
(3, 7)
6
6
4
4
2
2
y = x2 − 2
(−2, 2)
−4
−2
(−1, − 1)
Figure A.17
(2, 2)
x
2
(1, −1)
(0, −2)
4
−4
−2
Figure A.18
x
2
4
A18
Appendix A
Review of Elementary Algebra Topics
y
(−1, 3)
(−7, 3)
(− 2, 2)
Խ
5
Solution
Begin by creating a table of values, as shown below. Plot the solution points, as
shown in Figure A.19. It appears that the points lie in a “V-shaped” pattern, with
the point ͑Ϫ4, 0͒ lying at the bottom of the “V.” Following this pattern,
connect the points to form the graph shown in Figure A.20.
2
(− 3, 1) 1
(−5, 1)
x
− 7 − 6 −5 −4 −3 −2 −1
−1
(− 4, 0)
Խ
Sketch the graph of y ϭ x ϩ 4 .
4
3
(− 6, 2)
Sketching the Graph of an Equation
EXAMPLE 3
6
1
−2
x
Figure A.19
Խ
Ϫ7
Ϫ6
Ϫ5
Ϫ4
Ϫ3
Ϫ2
Ϫ1
3
2
1
0
1
2
3
Խ
y ؍x14
y
Solution
point
6
͑Ϫ7, 3͒ ͑Ϫ6, 2͒ ͑Ϫ5, 1͒ ͑Ϫ4, 0͒ ͑Ϫ3, 1͒ ͑Ϫ2, 2͒ ͑Ϫ1, 3͒
5
y= x+4
4
Intercepts of a graph are the points at which the graph intersects the x- or
y-axis. To find x-intercepts, let y ϭ 0 and solve the equation for x. To find
y-intercepts, let x ϭ 0 and solve the equation for y.
3
2
1
x
− 7 − 6 −5 −4 −3 −2 −1
−1
1
EXAMPLE 4
−2
Finding the Intercepts of a Graph
Find the intercepts and sketch the graph of y ϭ 3x ϩ 4.
Figure A.20
Solution
To find any x-intercepts, let y ϭ 0 and solve the resulting equation for x.
y
y-intercept:
(0, 4)
8
x-intercept:
4
6
(− 43 , 0)
Write original equation.
0 ϭ 3x ϩ 4
Let y ϭ 0.
4
Ϫ ϭx
3
y = 3x + 4
Solve equation for x.
To find any y-intercepts, let x ϭ 0 and solve the resulting equation for y.
x
y ϭ 3x ϩ 4
Write original equation.
−4
y ϭ 3͑0͒ ϩ 4
Let x ϭ 0.
−6
yϭ4
− 8 −6 −4
2
−8
Figure A.21
y ϭ 3x ϩ 4
4
6
8
Solve equation for y.
So, the x-intercept is ͑
0͒ and the y-intercept is ͑0, 4͒. To sketch the graph of
the equation, create a table of values (including intercepts), as shown below. Then
plot the points and connect them with a line, as shown in Figure A.21.
Ϫ 43,
x
Ϫ3
Ϫ2
Ϫ 43
Ϫ1
0
1
y ؍3x 1 4
Ϫ5
Ϫ2
0
1
4
7
͑Ϫ3, Ϫ5͒
͑Ϫ2, Ϫ2͒
͑0, 4͒
͑1, 7͒
Solution point
͑Ϫ 43, 0͒
͑Ϫ1, 1͒
Section A.4
A19
Graphs and Functions
Functions
A relation is any set of ordered pairs, which can be thought of as (input, output).
A function is a relation in which no two ordered pairs have the same first
component and different second components.
Testing Whether Relations Are Functions
EXAMPLE 5
Decide whether the relation represents a function.
a.
b. Input: 2, 5, 7
a
1
Output: 1, 2, 3
ͭ͑2, 1͒, ͑5, 2͒, ͑7, 3͒ͮ
2
b
3
c
4
Input
Output
Solution
a. This diagram does not represent a function. The first component a is paired
with two different second components, 1 and 2.
b. This set of ordered pairs does represent a function. No first component has two
different second components.
The graph of an equation represents y as a function of x if and only if no
vertical line intersects the graph more than once. This is called the Vertical Line Test.
Using the Vertical Line Test for Functions
EXAMPLE 6
Use the Vertical Line Test to determine whether y is a function of x.
y
y
a.
b.
3
3
2
2
1
1
x
x
−1
−1
1
2
3
4
5
−3 −2 −1
−1
−2
−2
−3
−3
1
2
3
Solution
a. From the graph, you can see that a vertical line intersects more than one point
on the graph. So, the relation does not represent y as a function of x.
b. From the graph, you can see that no vertical line intersects more than one point
on the graph. So, the relation does represent y as a function of x.
A20
Appendix A
Review of Elementary Algebra Topics
Slope and Linear Equations
The graph in Figure A.21 on page A18 is an example of a graph of a linear
equation. The equation is written in slope-intercept form, y ϭ mx ϩ b, where m
is the slope and ͑0, b͒ is the y-intercept. Linear equations can be written in other
forms, as shown below.
Forms of Linear Equations
1. General form: ax ϩ by ϩ c ϭ 0
2. Slope-intercept form: y ϭ mx ϩ b
3. Point-slope form: y Ϫ y1 ϭ m͑x Ϫ x1͒
The slope of a nonvertical line is the number of units the line rises or falls
vertically for each unit of horizontal change from left to right. To find the slope
m of the line through ͑x1, y1͒ and ͑x2, y2͒, use the following formula.
mϭ
y2 Ϫ y1 Change in y
ϭ
x2 Ϫ x1 Change in x
EXAMPLE 7
Find the slope of the line passing through ͑3, 1͒ and ͑Ϫ6, 0͒.
y
m=
3
1
9
2
(−6, 0)
9
−2
−3
Figure A.22
Finding the Slope of a Line Through Two Points
(3, 1)
1
x
Solution
Let ͑x1, y1͒ ϭ ͑3, 1͒ and ͑x2, y2͒ ϭ ͑Ϫ6, 0͒. The slope of the line through these
points is
4
mϭ
y2 Ϫ y1
0Ϫ1
Ϫ1 1
ϭ
ϭ
ϭ .
x2 Ϫ x1 Ϫ6 Ϫ 3 Ϫ9 9
The graph of the line is shown in Figure A.22.
You can make several generalizations about the slopes of lines.
Slope of a Line
1. A line with positive slope ͑m > 0͒ rises from left to right.
2. A line with negative slope ͑m < 0͒ falls from left to right.
3. A line with zero slope ͑m ϭ 0͒ is horizontal.
4. A line with undefined slope is vertical.
5. Parallel lines have equal slopes: m1 ϭ m2
6. Perpendicular lines have negative reciprocal slopes: m1 ϭ Ϫ
1
m2