Showing posts with label elementary algebra. Show all posts
Showing posts with label elementary algebra. Show all posts

Real Numbers and the Number Line

Natural Numbers – The set of counting numbers { 1, 2, 3, 4, 5, …}.
Whole Numbers – Natural numbers combined with zero  { 0, 1, 2, 3, 4, 5, …}.
Integers – Positive and negative whole numbers including zero {…,−5, −4, −3,−2, −1, 0, 1, 2, 3, 4, 5…}.
Rational Numbers – Any number of the form a/b where a and b are integers where b is not equal to zero.
Irrational Numbers – Numbers that cannot be written as a ratio of two integers.

When comparing real numbers, the larger number will always lie to the right of smaller numbers on a number line.  It is clear that 15 is greater than 5, but it might not be so clear to see that −5 is greater than −15.
Use inequalities to express order relationships between numbers.
<   "less than"
>   "greater than"
≤   "less than or equal to"
≥   "greater than or equal to"

It is easy to confuse the inequalities with larger negative values.  For example,
−120 < −10     “Negative 120 is less than negative 10.
Since −120 lies further left on the number line, that number is less than −10.  Similarly, zero is greater than any negative number because it lies further right on the number line.
0 > −59     "Zero is greater than negative 59."

Write the appropriate symbol, either < or >.
List three integers satisfying the given statement. (Answers may vary.)
Absolute Value – The distance between 0 and the real number a on the number line, denoted |a|. Because the absolute value is defined to be a distance, it will always be positive. It is worth noting that |0| = 0.
Point of confusion: You may encounter negative absolute values like this −|3|. Notice that the negative is in front of the absolute value. Work the absolute value first, then consider the opposite of the result. For example,
−|3| = −3
−|−7| = −7

Believe it or not, the above are correct! Look out for this type of question on an exam.

Video Examples on YouTube:

Distance and Midpoint Formulas

In this section we will review the distance and midpoint formulas.
Notice that the distance formula results in a real number and that the midpoint formula results in an ordered pair.

[ Interactive: Distance and Midpoint ]

Given two points find the distance and midpoint between them.

Video Lessons:
Use the distance and/or midpoint formulas to solve the following.

Example: If the diameter of a circle is defined by the two points (-3, 4) and (7, 4), find the center and radius of the circle. (Hint: diameter = 2r)
Example: Find the area of a circle given center (-3, 3) and point (3, 3) on the circle.
Video Examples on YouTube:

YouTube Example: Find the coordinates of an endpoint given the other endpoint and the midpoint.

Elementary Algebra (Algebra 1) Sample Exams

A complete set of sample exams covering topics found in the first 7 chapters of the online textbook Elementary Algebra.

You will find mobile friendly solutions as well as links to printable copies in pdf format. Please feel free to copy and paste anything you find here into your LMS.

Elementary Algebra Exam #3

Click on the 10 question exam covering topics in chapters 5 and 6. Give yourself one hour to try all of the problems and then come back and check your answers.

10. The length of a rectangle is 4 centimeters less than twice its width. The area is 96 square centimeters. Find the length and width. (Set up an algebraic equation then solve it)