Mathematical Relationships and Basic Graphs - Math

! is the symbol for factorial, which means the product of the whole numbers less than the given number.

Thus, .

By definition, .

An arithmetic sequence is one in which the common difference is added to one term to get the next term. Thus, if the first term is 3, we add 5 to get the second term, and continue in this manner.

Thus, the first four terms are:

A geometric sequence is one in which the next term is found by mutlplying the previous term by a particular constant. Thus, we look for an implicit definition which involves multiplication of the previous term. The only possibility is:

Begin by multiplying the numerator and denominator by the complement of the denominator:

Combine like terms:

Simplify. Remember that is equivalent to

Find the least common denominator (LCD) and convert each fraction to the LCD and then add. Simplify as necessary.

The result is an improper fraction because the numerator is larger than the denominator and can be simplified and converted to a mix numeral.

Every matrix has a dimension, which is represented as the number of rows and columns. For example, a matrix with three rows and two columns is said to have dimension 3 x 2.

The matrix

has two rows and three columns, so its dimension is 2 x 3. (Remember that rows go from left to right, while columns run up and down.)

Matrix multiplication is defined only if the number of columns in the first matrix is equal to the number of rows on the second matrix. The easiest way to determine this is to write the dimension of each matrix. For example, let's say that one matrix has dimension a x b, and the second matrix has dimension c x d. We can only multiply the first matrix by the second matrix if the values of b and c are equal. It doesn't matter what the values of a and d are, as long as b (the number of columns in the first matrix) matches c (the number of rows in the second matrix).

Let's go back to the problem and analyze the choice .

The dimension of the first matrix is 2 x 3, because it has two rows and three columns. The second matrix has dimension 2 x 2, because it has two rows and two columns.

We can't multiply these matrices because the number of columns in the first matrix (3) is not equal to the number of rows in the second matrix (2). Thus, this product is not defined.

The answer is .

Convert the mixed numbers into improper fractions by multiplying the whole number by the denominator and adding the numerator to get

Dividing by a fraction is the same as multiplying by its reciprocal so the problem becomes

Chenge the mixed numbers into improper fractions by multiplying the whole number by the denominator and adding the numerator to get

.

To find the -intercepts, we must set .

To solve absolute value equations, we must understand that the absoute value function makes a value positive. So when we are solving these problems, we must consider two scenarios, one where the value is positive and one where the value is negative.

Now we must set up our two scenarios:

and

and

and

Divide both sides by 3.

Consider both the negative and positive values for the absolute value term.

Subtract 2 from both sides to solve both scenarios for .

To solve absolute value equations, we must understand that the absoute value function makes a value positive. So when we are solving these problems, we must consider two scenarios, one where the value is positive and one where the value is negative.

and

This gives us:

and

However, this question has an outside of the absolute value expression, in this case . Thus, any negative value of will make the right side of the equation equal to a negative number, which cannot be true for an absolute value expression. Thus, is an extraneous solution, as cannot equal a negative number.

Our final solution is then

The absolute value measures the distance from zero to the given point.

In this case, since , or , as both values are twelve units away from zero.

The absolute value can never be negative, so the equation is ONLY valid at zero.

The equation to solve becomes .

Begin by multiplying the numerator and denominator by the complement of the denominator:

Combine like terms and simplify:

In finding the values for where , break the comparison of these two absolute value expressions into the four possible ways this could potentially be satisfied.

The first possibility is described by the inequality:

If you think of a number line, it is evident that there is no solution to this inequality since there will never be a case where subtracting from will lead to a greater number than adding to .

The second possibility, wherein is negative and converted to its opposite to being an absolute value expression but is positive and requires no conversion, can be represented by the inequality (where the sign is inverted due to multiplication by a negative):

We can simplify this inequality to find that satisfies the conditions where .

The third possibility can be represented by the following inequality (where the sign is inverted due to multiplication by a negative):

This is again simplified to and is redundant with the above inequality.

The final possibility is represented by the inequality

This inequality simplifies to . Rewriting this as makes it evident that this inequality is true of all real numbers. This does not provide any additional conditions on how to satisfy the original inequality.

The only possible condition that satisfies the inequality is that which arises in two of the tested cases, when .

The absolute value is the distance from a given number to 0. In our example, we are given -3. This number is 3 units away from 0, and thus the absolute value of -3 is 3.

If a number is negative, its absolute value will be the positive number with the same magnitude. If a number is positive, it will be its own absolute value.

The absolute value is a measure of the distance of a point from the origin. Using the pythagorean distance formula to calculate this distance.

Combine like terms. Treat as if it were any other variable.

Substitute to eliminate .

Simplify.