Download Unit 1 Review

Types of Numbers, The Number Line, Interval Notation, Absolute Value, Equations, and
Inequalities
Types of Numbers
Recall that the natural numbers are also called the counting numbers or the positive integers. These are the
numbers that you first learn about when you are a little kid, because these are the numbers that you count with:
1, 2, 3, 4, 5, etc.
When you get a little bit older, say in kindergarten, you also learn about the concept of zero, and then you have the
whole numbers which can also be thought of as the non-negative integers: 0, 1, 2, 3, 4, 5, ....
Somewhere around third grade the need for negative numbers arises. In life applications this can happen when
you owe money or play a card game where it is possible to go "in the hole". Indeed I first learned about negatives
by keeping score for my parents when they played cards and so when the topic came up in school, I said: "Oh, that
is the same thing as 'in the hole!'". I was very excited to see this connection. When you include the negatives of the
counting numbers along with the whole numbers, you have all of the integers: ... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ....
Fractions come into play when you learn to share, which is something that very small children don't do readily.
Sometimes when sharing objects amoung you and your friends, you don't have an amount that would allow each
person to have a whole number of the objects, so you figure out what fraction of it everyone is going to get. Also,
eventually, you start representing whole numbers on a number line and you realize that there is a lot of space
between two whole numbers which is plenty of room to add in the fractions which are more formally called the
rational numbers. Rational numbers can be written as a fraction of two integers where the bottom integer is not
zero. They can also be written as a decimal, but something special happens when you write them as a decimal. What
is special about the decimal version of a rational number?
Try it with:
1. 7/8,
2. 3/5,
3. 2/3, and
4. 1/7.
5. What properties did these rational numbers all have when written as a decimal?
What you don't realize yet, is that there are still alot of holes in that number line. Indeed it took along time and a lot
of persacution for humans to realize that there was a need for numbers that are not fractions of integers. A couple of
the earliest non-rational numbers that came up, were arrived at when the hypotenuese of a right triangle was being
calculated, and the ratio of the circumference of a circle divided by its diameter. Can you show that these numbers
are not simple fractions? How do we know that numbers like √2 or π don't fit into the category of rational numbers
listed above? When we include these kind of numbers, we are talking about the irrational numbers. They are
irrational, because they cannot be written as a ratio or fraction of two integers. Irrational numbers can be plotted on
the number line and they fill in the wholes that were left by the rational numbers. Can you find a rational and an
irrational number between any two rational numbers and irrational numbers?
Now we need some kind of set that includes both rational and irrational numbers. The set of all numbers that we
can plot on the number line: the rationals and the irrationals is called the real numbers.
Come up with a rational number and an irrational number that is between:
6. 31/64 and 37/75
7. 0.435 and 0.435001
You may want to use a calculator to assist you with these two problems.
Properties of Numbers:
associative: (ab)c = a(bc) or a + (b + c) = (a + b) + c,
commutative: ab = ba or a + b = b + a
and distributative: a(b + c) = ab + ac.
Interval Notation and One-Diminsional Graphs
Simplify and graph on a number line:
8. (-∞, 2)∩{[-3,-1)∪(1.5, ∞)}
9. {(-∞,2)∪[-3, -1)}∩{(-∞, 2)∪(1.5, ∞)}
Absolute Value Definition
The absolute value of a number can be thought of as its positive value or as its distance from the origin (from zero)
when the number is plotted on the number line. Distance is always positive. Vertical bars are used to represent
absolute value.
Examples:
|5| = 5
|-4| = 4
Given x < 0, |x| = -x
Given x > 0, |x| = x
Evaluate or simplify:
10. |4 - 10| = _____
11. |10 - 4| = _____
12. given x > 3, |x - 3| = _______
13. given x > 5, |5 - x| = _______
Absolute Value as a Distance
Example
"The distance from a number to 4 is at least 7" can be written as below. |x - 4| ≥ 7 and illustrated as
14. Rewrite "the distance from a number to -3 is no more than 2" as an absolute value involving distance and
represent this on a graph.
15. Write |x - 8| > 3 as a statement involving distance and graph it.
16. Look at the following graph and write a statement involving distance and an inequality involving absolute
value that is equivalent to this graph.
Solving Linear Equations for One Variable
When solving linear equations, if the variable that you are solving for only appears once in the equation, then you
can simply do the opposite of what ever is being done to it, to move everything to the other side of the equation so
you can isolate the variable. If the variable appears more than once, then first combine like terms before isolating
the variable.
Examples:
3x = 8(3 - 5x)
3x = 24 - 40x
43x = 24
x = 24/43
Solve for a: wa = ta + b
wa - ta = b
a(w - t) = b
a = b/(w - t)
Sove each of the following expressions for the variable.
17. 7(x + 20) -13 = 19
18. 5x - 6(x + 3) = 2x + 12
19. P = 2w + 2l for w
Solving Absolute Value Equations
Isolate the absolute value first, then translate it into two equivalent equations, and then solve each equation.
Example:
-3|2x - 7| + 20 = 15
-3|2x - 7| = -5
|2x - 7| = 5/3
2x - 7 = 5/3 or 2x - 7 = -5/3
2x = 26/3 or 2x = 16/3
x = 13/3 or x = 8/3
20. Solve: |3 + 5x| - 2 = 9
Solving Linear Inequalitities
Examples: Solve for x and write your answer in interval notation if appropriate.
7(x + 3) < 14
7x + 21 < 14
7x <-7
x < -1
(-∞, -1)
4x ≥ 10x + 12
-6x ≥ 12
x ≤ -2 (Recall that multiplying an inequality by a negatie flips the direction of the inequality.)
(-∞, -2]
Solve each of the following and write your answer in interval notation.
21. 5(x - 7) + 3x ≤ 8(x + 2)
22. -12(x - 4) > 39
Solving Compound Inequalities
Sometimes you need to deal with more than one inequality. Recall that union symbolized by &cup means the same
thing as 'or' and includes all answers that are in the solution set of at least one inequality. Also, intersection
symbolized by ∩ means the same thing as 'and' and includes only answers that are in all of the sets that you are
intersecting. This makes sense if you think about the intersection of two streets only being the part where the streets
cross each other. Your final answer should never contain an intersection symbol since those can always be
simplified, but sometimes you will need a union symbol if you have two intervals that don't overlap.
Examples:
2x + 5 ≥ 7 and -3x + 6 > 0
2x ≥ 2 ∩ -3x > -6
x≥1 ∩ x<2
1≤x<2
x ∈ [1, 2)
2x + 5 ≥ 7 or -3x + 6 > 0
x≥1∪x<2
x ∈ (-∞, ∞)
x > 3 and -x ≥ 2
x >3 and x ≤ -2
no solution
Solve each of the following and write your answer in interval notation.
23. 2(5x + 7) > 10 ∩ 7x < 28
24. x < 3x + 6 or 4x + 12 < 0
25. x ≥ 3x + 6 and 4x + 12 ≥ 0
Solving Compound Inequalitites Involving Absolute Value
Now, put all of the ideas together. Use a number line graph if needed to help you figure it out.
Example:
|x + 3| < 4 and |x + 3| ≥ 2
-4 < x + 3 < 4 and (x + 3 ≤ -2 or x + 3 ≥ 2)
-7 < x < 1 and (x ≤ -5 or x ≥ -1)
(-7, -5]∪[-1,1)
You try:
26. |3 - x| > 4 and |x - 2| ≤ 6
27. |2x - 5| ≤ 12 ∩ |2x - 11| ≥ 6
28. |2x - 5| > 12 ∪ |2x - 11| < 6
Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
0.875
0.6
0.6
0.142857
They either repeated or terminated.
Answers will vary: 31/64 = 0.4843375 and 37/75=0.493, so a rational example would be: 0.49 and an
irrational example would be: √(62)/16
Answers will vary. A rational example would be 0.4350005 and an irrational example would be
0.4350001000010000010000001...
[-3, -1)∪(1.5, 2)
(-∞, 2) (graph image to be added later)
6
6
x-3
x - 5 (graph image to be added later)
|x + 3| ≤ 2 (graph image to be added later)
The distance from a number to 8 is more than 3. (graph image to be added later)
The distance from a number to 12 is less than 4. |x - 12| < 4
-108/7
x = -10
P/2 - l
8/5 or -14/5
(-∞, ∞)
(-∞, 3/4)
(-2/4, 4)
(∞, -3)∪(-3, ∞)
{-3}
[-4, -1)∪(7, 8]
[-3.5, 2.5]∪{8.5}
(∞, -3.5)∪(2.5, 8.5)∪(8.5, ∞)