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:

Solving Equations Quadratic in Form

In this section, we make use of all the techniques that we have learned so far for solving quadratic equations. In fact, the equations found here are reducible to quadratic form.

YouTube U-sub Solving Playlist

Here are most of the reducible equations that we are likely to encounter. Begin by trying to identify what can be squared to obtain the leading variable term.

Tip: Look at the middle term for a hint as to what u should be.

In the previous solved problem, we certainly could have distributed the expression on the left side, put the equation in standard form then re-factored it. Instead, here we are illustrating a technique that will be used to easily solve many other equations that are quadratic in form.

Solve by making a u-substitution.

 

 

 

Solve: x^6 + 26x^3 -27

 

Six Answers: { -3, 1, (3±3iSqrt(3))/2, (-1±iSqrt(3))/2 }

So far we have been able to factor after we make the u-substitution.  If the resulting quadratic equation does not factor, then use the quadratic formula.

YouTube Videos:

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