MATH
JohnStud
2.2.docxIntroduction
Course Objectives
This lesson will address the following course outcomes:
· 15. Solve linear equations in one variable, including problems involving the distributive property and fractions.
· 16. Construct inequalities to represent relationships, solve simple and compound inequalities in one variable, represent solutions using interval notation, and interpret solutions in the context of the situation.
Specific Objectives
Students will understand that
· an inequality represents a range of values.
· a compound inequality represents a range of numbers between two end values.
· solutions to inequalities are a range of numbers.
· solving inequalities is very similar to solving equations (with one important difference.)
Students will be able to
· solve linear equations in one variable.
· construct inequalities to represent real-world relationships.
· solve simple and compound inequalities.
· represent solutions using interval and number-line notations.
Equations and Inequalities
In the last lesson, you worked with expressions, which consist of numbers and variables, but do not contain an equal sign. In this lesson we will focus on equations, which always include an equal sign, and inequalities, which are like an equation but involves an inequality sign.
Usually with equations what we need to do is solve the equation. When the equation involves one variable, we solve for the value of that variable that makes the equation true. When the equation involves more than one variable, we solve for one variable in terms of the others, so we can easily calculate the value of that variable when we know values for the others.
Solving Equations
Equations can be solved by using the properties of equality, which states that whatever you do to one side of an equation, you must do to the other side to maintain the equality.
|
Additive Property of equality (works for subtraction too) |
If A = B, then A + C = B + C If A = B, then A – C = B – C |
|
Multiplicative Property of equality (works for division too) |
If A = B, then AC = BC If A = B, then A/C = B/C (C ≠ 0) |
Example: Solve 3x+4=193x+4=19
|
Subtract 4 from both sides: |
3x+4−4=19−43x+4-4=19-4 3x=153x=15 |
|
Divide both sides by 3: |
3x3=1533x3=153 x=5x=5 Sometimes we need to simplify before solving. |
For some more examples of solving linear equations, watch these videos:
· Solving Two-Step Equations [+]
· Solving Multi-Step Equations [+]
In the next three questions, provide your answers as fractions or integers - do not give decimal answers.
Applying Solving
In the last lesson, you came up with a formula for the tax a couple owed based on their wage income. A similar formula for a single person with taxable income more than $9,075 but not over $36,900 is
T=0.15w−1976.25T=0.15w-1976.25 , where T is the tax owed, and w is the wage income.
This formula is an example of a linear equation, and equation where every term is a constant or a single variable not raised to a power. This particular linear equation involves two variables. In this equation w would be considered the input or independent variable since it is the value typically provided to the equation, and T is the output or dependent variable since its value depends on the input, and its value is the output of the expression on the right side of the equation.
Inequalities
A difficulty with our equation is that the input is wage income, but the formula gives a range of taxable incomes it works for. This is confusing. To rectify this, we need to talk about inequalities. Inequalities can be used for comparisons, and to represent sets of values.
Inequalities Representing Comparisons
· The inequality 2 < 7 is a comparison, stating the 2 is smaller than 7. Likewise, 4 > -8 means 4 is greater than -8.
· If the variable S represented the numbers of hours you spent studying last week, and G represented the number of hours you spent playing video games last week, then the inequality S ≥ G would say that the number of hours you spent studying was equal to or greater than the number of hours you spent playing video games.
· In common language, this would probably get stated equivalently as "you spent the same or more hours studying compared to hours playing video games" or "you spent at least as many hours studying as you did playing video games".
Inequalities Representing Sets of Values
Inequalities can also be used to represent a set of values
· For example, suppose an employer pays benefits for anyone who works 20 or more hours a week. If we let h represent the number of hours a week someone works, then we could represent the people who earn benefits using the inequality h ≥ 20. Graphically this could be illustrated as an interval on a number line:
· To represent a set of values between two bounds, a compound inequality is used. For example, the statement "this tax credit is available for people making from $25,000 up to $45,000" could be represented by the compound inequality 25,000 ≤ x ≤ 45,000, where x represents income. Graphically this is illustrated as an interval on the number line that includes both endpoints:
· When writing compound inequalities, it is most common to write them with less-than or less-than-or-equal symbols. It is essential that the value on the left also be less than the value on the right. For example, 2 < x < 7 is a valid compound inequality, but 5 < x < 2 is not, because 5 is not less than 2, so it is impossible for x to be both larger than 5 and less than 2.
Interval notation is another way to represent an inequality representing a set or range of values.
· For example, 2 < x < 7 in interval notation is written as (2 , 7). The parentheses indicate that the 2 and 7 are not included in the interval.
· If the endpoints were included, then square brackets would be used rather than parentheses. So: 2 < x < 7 in interval notation is written as [2 , 7]
Examples of sets of values, represented in words, inequalities, interval notation, and graphs:
· All of the numbers between 0 and 3
Inequality: 0 < x < 3 Interval: (0, 3)
· All of the numbers between 0 and 3 and including 0 and 3
Inequality: 0 < x < 3 Interval: [0, 3]
· All of the numbers greater than 1. Notice that the interval notation uses the infinity symbol, ∞∞ .
Inequality: x > 1 Interval: (1, ∞)
· All of the real numbers.
Interval: (-∞, ∞)
· Your report must be more than 3 pages long but must be a max of 10 pages.
Inequality: 3 < n < 10 Interval: (3, 10]
· A grade of “C” in the course is earned when a student has a grade average, g, from .75 (75%) up to but not including .85 (85%)
Inequality: 0.75 < g < 0.85 Interval: [.75, .85)
Note: When the infinity sign ∞ is used in interval notation, a parenthesis is always placed next to it, not a square bracket, because there is no ending value
Comparison of Expressions, Equations, and Inequalities
Look at the table below for some examples of how expressions, equations, and inequalities compare.
|
Symbol |
Meaning |
Comments |
|
xx |
x represents some number. |
This is a variable. We don’t know what that number is. |
|
x=8x=8 |
x equals 8 |
This is an equation. We are certain that x is equal to 8 |
|
x<8x<8 |
x represents all those numbers less than 8. |
x can be any number less than 8. |
|
|
|
|
|
x+5x+5 |
Represents the idea that I am adding 5 to whatever x is. |
This is called an expression |
|
x+5=8x+5=8 |
x is some number that when I add 5 to it, I get 8. |
This is called an equation. If I add 5 to the number 3, I get 8 so if I solve the equation, x=3x=3 |
|
x+5<8x+5<8 |
x is some number that when I add 5 to it, I get something less than 8. |
Any number less than 3 can be added to 5, the sum will still be less than 8, so if I solve the inequality, x<3x<3 |
Solving Inequalities
Solving Equations and Inequalities
Here are some examples of solving equations and inequalities.
|
EQUATION |
INEQUALITY |
COMPOUND INEQUALITY |
|
x+5=8x+5=8 x+5−5=8−5x+5-5=8-5 x=3x=3
Check: 3 + 5 = 8 TRUE |
x+5<8x+5<8 x+5−5<8−5x+5-5<8-5 x<3x<3
In interval notation: (-∞, 3) Check: For example, pick 1 for x then 1 + 5 < 8 TRUE Or pick – 4 for x; – 4 + 5 < 8
Pick 2.9 for x; 2.9 + 5 < 8 Pick 2.99 for x; 2.99 + 5 < 8 Pick 4 for x; 4 + 5 < 8 NOT TRUE |
1<x+t≤81<x+t≤8 1−5<x+5−5≤8−51-5<x+5-5≤8-5 −4<x≤3-4<x≤3
In interval notation: (-4, 3] Check: For example, pick 2 for x, then 1 < 2 + 5 ≤ 8 TRUE
Or pick – 4 for x, then 1 < – 4 + 5 ≤ 8 NOT TRUE Pick 4 for x, then 1 < 4 + 5 ≤ 8 NOT TRUE |
When we solve inequalities, you use the same rules as for equations, except:
· When you multiply or divide each side of the inequality by a negative number, you must change the direction of the inequality symbol.
For more about inequalities, watch these videos:
· Introduction to Basic Inequalities in One Variable [+]
· Solving Linear Inequalities in One Variable [+]
· Introduction to Basic Compound Inequalities [+]
· Ex 1: Solve a Compound Inequality [+]
· Ex 2: Solve a Compound Inequality [+]
#1 Points possible: 5. Total attempts: 5
Solve for x: 3x+4=−83x+4=-8
x =
#2 Points possible: 5. Total attempts: 5
Solve for x: −2x+6=5x+15-2x+6=5x+15
x =
#3 Points possible: 5. Total attempts: 5
Solve for x: 4x+5−5x+6=5−3(x−4)4x+5-5x+6=5-3(x-4)
x =
#4 Points possible: 5. Total attempts: 5
Using the formula provided, find the tax owed by someone with wage income of $30,000.
$
#5 Points possible: 5. Total attempts: 5
Using the formula provided, find the wage income of someone who owes $3,423.75 in taxes.
$
#6 Points possible: 8. Total attempts: 5
Entering Interval and Inequality Answers
For intervals of values, some questions will ask for your answer as an inequality, and others will ask for your answer in interval notation.
There are two ways to enter these types of answers:
1. Enter using calculator-style notation (see below)
2. After clicking in the answer box, click the yellow arrow that shows to the right of it to use the MathQuill equation editor. Click the "Intervals" or "Inequalities" tab. Watch a demo [+]
Here are some examples of how interval notation relates to inequalities, and how you'd enter them in calculator-style notation.
|
Inequality |
To enter this, type |
Interval Notation |
To enter this, type |
|
3<x<53<x<5 |
3 < x < 5 |
(3,5) |
(3,5) |
|
3<x≤53<x≤5 |
3 < x <= 5 |
(3,5] |
(3,5] |
|
x>3x>3 |
x > 3 |
(3,∞)(3,∞) |
(3,oo) |
|
All real numbers |
All real numbers |
(−∞,∞)(-∞,∞) |
(-oo,oo) |
When entering intervals, use round parentheses for "less/great than", and use square brackets for "less/greater than or equal to". Use oo (two lowercase letter o's) for infinity: ∞∞
Try it now:
a) Enter [3,∞)[3,∞) in interval notation.
b) Enter 2<x≤72<x≤7 in inequality notation.
#7 Points possible: 5. Total attempts: 5
Solve the equation: 2x+5=82x+5=8
x =
#8 Points possible: 12. Total attempts: 5
Solve the inequality: 2x+5≤82x+5≤8
The answer as an inequality:
The answer in interval notation:
The answer on a number line:
Clear All Draw: Line SegmentDotOpen Dot
#9 Points possible: 12. Total attempts: 5
Solve the inequality: 1≤2x+5<81≤2x+5<8
The answer as an inequality:
The answer in interval notation:
The answer on a number line:
Clear All Draw: Line SegmentDotOpen Dot
#10 Points possible: 12. Total attempts: 5
Solve the inequality: x+5>8x+5>8
The answer as an inequality:
The answer in interval notation:
The answer on a number line:
Clear All Draw: Line SegmentDotOpen Dot
#11 Points possible: 5. Total attempts: 5
Solve the equation: −2x+5=8-2x+5=8
x =
#12 Points possible: 8. Total attempts: 5
Solve the inequality: −2x+5<8-2x+5<8
The answer as an inequality:
The answer in interval notation:
#13 Points possible: 8. Total attempts: 5
Solve the inequality: 1<−2x+5<81<-2x+5<8
The answer as an inequality:
The answer in interval notation:
#14 Points possible: 5. Total attempts: 5
Think back to our tax calculation formula. We know the formula we used was valid when 9.075<x≤36,9009.075<x≤36,900, where x represents the taxable income. For a single person with only wage income, the taxable income is wage income minus a $10,150 deduction, so x=w−10,150x=w-10,150 .
Substituting this into the inequality above gives 9,075<w−10,150≤36,9009,075<w-10,150≤36,900. Solve this inequality.
