Inequalities and Linear Programming - Pre-Calculus

When we work with absolute value equations, we're actually solving two equations:

and

Adding to both sides leaves us with:

and

Dividing by in order to solve for allows us to reach our solution:

and

Which can be rewritten as:

In order to solve for we must first isolate the absolute value. In this case, we do it by dividing both sides by 2:

As with every absolute value problem, we set up our two equations:

and

We isolate by adding to both sides:

and

Finally, we divide by :

and

Our first step in solving this equation is to isolate the absolute value. We do this by dividing both sides by

.

We then set up our two equations:

and .

Subtracting 4 from both sides leaves us with

and .

Lastly, we multiply both sides by 2, leaving us with :

and .

Which can be rewritten as:

We first need to isolate the absolute value, which we can do in two steps:

1. Add 2 to both sides:

2. Divide both sides by 4:

Our next step is to set up our two equations:

and

We can now solve the equations for by subtracting both sides by 8:

and

and then dividing them by 5:

and

Which can be rewritten as:

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by subtracting seven from both sides.

Next we need to set up two inequalities since the absolute value sign will make both a negative value and a positive value positive.

From here, subtract thirteen from both sides and then divide everything by four.

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by dividing both sides by three.

We now have two equations:

and

So, our solution is

First we need to get the expression with the absolute value sign by itself on one side of the inequality. We can do this by subtracting two from both sides then dividing everything by three.

Since absolute value signs make both negative and positive values positive we need to set up a double inequality.

Now to solve for subtract four from each side.

If , then either or based on the meaning of the absolute value function. We have to solve for both cases.

a) subtract 5 from both sides

divide by -2, which will flip the direction of the inequality

Even if we didn't know the rule about flipping the inequality, this answer makes sense - for example, , and .

b) subtract 5 from both sides

divide by -2, once again flipping the direction of the inequality

First, simplify so that the absolute value function is by itself on one side of the inequality.

.

Note that the symbol flipps when you divide both sides by .

Next, the two inequalities that result after removing the absolute value symbols are

and .

When you simplify the two inequalities, you get

and .

Thus, the solution is

.

To solve absolute value inequalities, you have to write it two different ways. But first, divide out the 4 on both sides so that there is just the absolute value on the left side. Then, write it normally, as you see it: and then flip the side and make the right side negative: . Then, solve each one. Your answers are and .

To solve absolute value inequalities create two functions.

Simply remember that when solving inequalities with absolute values, you keep one the same and then flip the sign and inequality on the other.

Thus,

Then, you must write it in interval notation.

Thus,

1. Write as two simple inequalities:

1. Solve the inequalities:

1. Write the final solution as a single compound inequality:

For interval notation:

1. Now graph:

First, we must solve for the roots of the cubic polynomial equation.

We obtain that the roots are .

Now there are four regions created by these numbers:

• . In this region, the values of the polynomial are negative (i.e.plug in and you obtain

• . In this region, the values of the polynomial are positive (when , polynomial evaluates to )

• . In this region the polynomial switches again to negative.

• . In this region the values of the polynomial are positive

Hence the two regions we want are and .

1. Multiply both sides of the equation by the common denominator of the fractions:

1. Simplify:

1. For standard notation, and the fact that inequalities can be read backwards:

For interval notation:

1. Graph:

Graph the rational expression,

1. Because and a divide by is undefined in the real number system, there is a vertical asymptote where .

2. As , , and as , .

3. As , , and as , .

4. The funtion y is exists over the allowed x-intervals:

One approach for solving the inequality:

For

1. Determine where over the x-values or .

2. for the intervals or .

3. Then the solution is .

Another approach for solving the inequality:

1. Write as , then determine the x-values that cause to be true:

1. is true for or .

2. Then the solution is .

First, subtract from both sides so you get

.

Then find the common denominator and simplify

.

Next, factor out the numerator

and set each of the three factor equal to zero and solve for .

The solutions are

.

Now plug in values between , , , and into the inequality and observe if the conditions of the inequality are met.

Note that . They are met in the interval and .

Thus, the solution to the inequality is

.

The zeros of the function are the values of where the function will be equal to zero. In order to find these we set the numerator of the function equal to zero.

We only need to solve for once,

So the zeros of this function are .

To solve for the points at which this function will be undefined, we set the denominator equal to zero and solve for .

And so the function is undefined at

We begin by finding the zeros of the equation using the numerator.

So we know that the function will equal zero when . If we just look at the numerator of the function, then this graph would be a parabola with its point at . Now we will solve for the points where the function is undefined by setting the denominator equal to zero and solving for .

And so the function is undefined at . If we make a table to solve for some of the points of the graph:

x	y

And if we graph these points we see something like below (which is our answer). Note that the dotted blue line is the vertical asymptote at .

When solving for a point where the function will be undefined, you set the denominator equal to zero and solve for . This creates a vertical asymptote because when the denominator equals zero the function is undefined and we are solving for . Say for example a function is undefined at . So at all values of where this function is undefined creating a vertical asymptote.