Solve a quadratic inequality by first putting it in the standard form  (or similar); this is done by subtracting  from both sides:

Factor the quadratic trinomial on the left:

We are looking for two integers whose product is 32 and whose sum is . Through a little trial and error, we find  and , so

,

making the inequality 

The boundary points of the solution set are the points at which either factor is equal to 0:

, in which case , and

, in which case .

These points divide the real numbers into three intervals: 

We can choose one test point from each interval to determine the truth of each statement on the interval:

: choose  

True: include 

: choose 

False: exclude .

: choose 

True: include 

The boundaries are also included, since the inequality allows for inclusion ("greater than or equal to"). The solution set is .

The boundary points of a rational inequality are those values of the variable for which the numerator or the denominator are equal to 0:

If numerator

, then

.

Since equality is permitted in the statement (  ), include this value of .

If denominator 

,

then 

Since this value makes the denominator 0, exclude it, regardless of the inequality symbol.

There are three intervals which must be tested for inclusion or exclusion:

, , 

Test one value on each interval in the original inequality for truth or falsity in order to determine which intervals should be included:

: Set 

False; exclude .

: Set 

True; include 

: Set 

False; exclude 

Since  is excluded as a solution and 4 is included, the correct solution set is the interval .

Solve a quadratic inequality by first putting it in the standard form  (or similar); this is done by first applying the FOIL method to the product of the binomials on the right:

Subtract 21 from both sides:

The  method can be used to factor the trinomial; we are looking for two integers whose product is  and whose sum is ; by trial and error, we find that the numbers are 7 and . The trinomial can be rewritten:

Factoring by grouping:

The boundary points of the solution set are the points at which either factor is equal to 0:

and 

These points divide the real numbers into three intervals: 

Test one value on each interval in the original inequality for truth or falsity in order to determine which intervals should be included:

: Set 

True: include .

: Set 

False: exclude 

: Set 

True: include .

The boundaries are also included, since the inequality allows for inclusion ("greater than or equal to"). The solution set is .

Get all expressions in the rational inequality over to the same side by adding 4 to both sides:

Rewrite the expression on the left as a single rational expression, as follows:

The boundary points of a rational inequality are those values of the variable for which the numerator or the denominator are equal to 0:

If numerator

then

Since equality is permitted in the statement (  ), include this value of .

If denominator 

then

.

Since this value makes the denominator 0, exclude it, regardless of the inequality symbol.

There are three intervals which must be tested for inclusion or exclusion:

, , 

Test one value on each interval in the original inequality for truth or falsity in order to determine which intervals should be included:

: Set 

True; include 

: Set 

False: exclude 

: Set 

True: include .

Since  is excluded as a solution and  is included, the correct solution set is 

In an absolute value inequality, the absolute value expression must be isolated on one side first. We can do this by subtracting 21 from both sides:

This can be rewritten as the three-part inequality

Subtract 14 from all three expressions:

Divide all three expressions by , reversing the inequality symbols since you are dividing by a negative number:

In interval notation, this is .

In an absolute value inequality, the absolute value expression must be isolated on one side first. We can di this by first subtracting 42 from both sides:

Divide by , reversing the direction of the inequality symbol since we are dividing by a negative number:

This inequality can be rewritten as the compound inequality

 or 

Solve each simple inequality separately. 

Subtract 19 from both sides:

Divide by , remembering to reverse the symbol:

In interval notation, this is .

Carry out the same steps on the other simple inequality:

In interval notation, this is .

Since the two simple inequalities are connected by an "or", their individual solution sets are connected by a union; the solution set is

.

In general, an equation of depreciation looks like the following.

, where  is the starting amount,  is the common ratio, and  is time.

For us , , and .

To answer this question, we should create a table of values and compare. For calculating the difference between the y value and the y value of the line of best, take the absolute value since it won't make since if it is negative.

From our table of values, we can see that the point furthest from the line of best fit is .

First step is to set it equal to zero

Now factor out an 

Set up the equations, and solve for .