Inequalities and Linear Programming - Pre-Calculus

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. 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 .

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 this equation, we must first isolate the absolute value. In this case, we do it by dividing both sides by which leaves us with:

When we work with absolute value equations, we're actually solving two equations. So, our next step is to set up these two equations:

and

In both cases we solve for by adding to both sides, leaving us with

and

This 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:

In order to solve a system of linear equations, we must start by solving one of the equations for a single variable:

We can now substitute this value for y into the other equation and solve for x:

Our last step is to plug this value of x into either equation to find y:

We first move to the left side of the equation:

Subtract the bottom equation from the top one:

Left Side:

Right Side:

So

So dividing by a -1 we get our result.

For any system of linear equations, we can start by solving one equation for one of the variables, and then plug its value into the other equation. In this system, however, we can see that both equations are equal to y, so we can set them equal to each other:

Now we can plug this value for x back into either equation to solve for y:

So the solutions to the system, where the lines intersect, is at the following point:

In order to solve a system of linear equations, we can either solve one equation for one of the variables, and then substitute its value into the other equation, or we can solve both equations for the same variable so that we can set them equal to each other. Let's solve both equations for y so that we can set them equal to each other:

Now we just plug our value for x back into either equation to find y:

So the solution to the system is the point: