Intermediate Algebra Tutorial 17

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When I'm working with only the boundary line, I will put an equal sign between the two sides to emphasize that we are working on the boundary line.  That doesn't mean that I changed the problem. When we put it all together in the end, I will put the inequality back in.

What value is y on the x-intercept?
If you said 0, you are correct.
If you need a review on x-intercepts, go to Tutorial 14: Graphing Linear Equations.
 

*x-intercept
 

Since the x-intercept came out to be (0, 0), then it stands to reason that when we put in 0 for x to find the y-intercept, we will get (0, 0).
 

Let's move on and plug in 1 for x to get a second solution:
 

 

Plug in -1 for x to get a third solution:
 

 

Solutions:
  x y (x, y) 0 0 (0, 0) 1 -3 (1, -3) -1 3 (-1, 3)

Since the original problem has a >, this means it DOES include the boundary line. 

So are we going to draw a solid or a dashed line for this problem? 
It looks like it will have to be a solid line.

Putting it all together, we get the following boundary line for this problem:

 

 

An easy test point would be (1, 1).  Note that it is a point that is not on the boundary line. In fact, it is located above the boundary line. 

Let's put (1, 1) into the original problem and see what happens:
 

 
 

Our solution would lie above the boundary line.  This means we will shade in the part that is above it.

Note that the gray lines indicate where you would shade your final answer.

(return to problem 1a)
 
 

 

 

 

Do you remember what type of line y = c graphs as?
It comes out to be a horizontal line. 
If you need a review on horizontal lines, go to Tutorial 14: Graphing Linear Equations.

Every y's value on the boundary line would have to be 3. 

Solutions:
  x y (x, y) 0 3 (0, 3) 1 3 (1, 3) 2 3 (2, 3)
 

So are we going to draw a solid or a dashed line for this problem? 
It looks like it will have to be a dashed line.

Putting it all together, we get the following boundary line for this problem:

 

 

An easy test point would be (0, 0).  Note that it is a point that is not on the boundary line. In fact, it is located below the boundary line. 

Let's put (0, 0) into the original problem and see what happens:
 

 
 

Our solution would lie below the boundary line.  This means we will shade in the part that is below it

Note that the gray lines indicate where you would shade your final answer.

(return to problem 1b)
 
 

 

  OR 
 

Step 1:  Graph the boundary line.
 

Do you remember what type of line y = c graphs as?
It comes out to be a horizontal line. 
If you need a review on horizontal lines, go to Tutorial 14: Graphing Linear Equations.

Every y's value on the boundary line would have to be -5. 

Solutions:
  x y (x, y) 0 -5 (0, -5) 1 -5 (1, -5) 2 -5 (2, -5)
 

So are we going to draw a solid or a dashed line for this problem? 
It looks like it will have to be a solid line.

Putting it all together, we get the following boundary line for this problem:

 

 

An easy test point would be (0, 0).  Note that it is a point that is not on the boundary line. In fact, it is located above the boundary line. 

Let's put (0, 0) into the original problem and see what happens:
 

 
 

Our solution would lie above the boundary line.  This means we will shade in the part that is above it.

Note that the gray lines indicate where you would shade your final answer.

 
 
 

Step 1:  Graph the boundary line.
 

Do you remember what type of line x = c graphs as?
It comes out to be a vertical line. 
If you need a review on vertical lines, go to Tutorial 14: Graphing Linear Equations.

Every x's value on the boundary line would have to be 2. 

Solutions:
  x y (x, y) 2 0 (2, 0) 2 1 (2, 1) 2 2 (2, 2)
 

So are we going to draw a solid or a dashed line for this problem? 
It looks like it will have to be a dashed line.

Putting it all together, we get the following boundary line for this problem:

 

 

An easy test point would be (0, 0).  Note that it is a point that is not on the boundary line. In fact, it is located to the left of the boundary line. 

Let's put (0, 0) into the original problem and see what happens:
 

 
 

Our solution would lie to the right of the boundary line.  This means we will shade in the part that is to the right of it

Note that the gray lines indicate where you would shade your final answer.

 
 

This is what we get when we union these two inequalities:

Note that the gray lines indicate where you would shade your final answer.

(return to problem 1c)
 
 

 

  AND 
 
 

Step 1:  Graph the boundary line.
 

When I'm working with only the boundary line, I will put an equal sign between the two sides to emphasize that we are working on the boundary line.  That doesn't mean that I changed the problem. When we put it all together in the end, I will put the inequality back in.

What value is y on the x-intercept?
If you said 0, you are correct.
If you need a review on x-intercepts, go to Tutorial 14: Graphing Linear Equations.
 

*Inverse of mult. by 2 is div. by 2
 

*x-intercept
 

What is the value of x on the y-intercept?
If you said 0, you are correct.
If you need a review on y-intercepts, go to Tutorial 14: Graphing Linear Equations.
 

*Inverse of mult. by 2 is div. by 2
 

*y-intercept
 

Plug in 1 for x to get a third solution:
 

*Inverse of add 2 is sub. 2

*Inverse of mult. by 2 is div. by 2
 
 

 
 

Solutions:
  x y (x, y) 2 0 (2, 0) 0 2 (0, 2) 1 1 (1, 1)

Since the original problem has a >, this means it DOES NOT include the boundary line. 

So are we going to draw a solid or a dashed line for this problem? 
It looks like it will have to be a dashed line.

Putting it all together, we get the following boundary line for this problem:

 

 

An easy test point would be (0, 0).  Note that it is a point that is not on the boundary line. In fact, it is located below the boundary line. 

Let's put (0, 0) into the original problem and see what happens:
 

 
 

Our solution lies below the boundary line.  This means we will shade in the part that is below it.

Note that the gray lines indicate where you would shade your final answer.

 
 
 

Step 1:  Graph the boundary line.
 

When I'm working with only the boundary line, I will put an equal sign between the two sides to emphasize that we are working on the boundary line.  That doesn't mean that I changed the problem. When we put it all together in the end, I will put the inequality back in.

What value is y on the x-intercept?
If you said 0, you are correct.
If you need a review on x-intercepts, go to Tutorial 14: Graphing Linear Equations.
 

*Inverse of mult. by 2 is div. by 2

*x-intercept
 

What is the value of x on the y-intercept?
If you said 0, you are correct.
If you need a review on y-intercepts, go to Tutorial 14: Graphing Linear Equations.
 

*Inverse of mult. by -4 is div. by -4
 

*y-intercept
 

Plug in 1 for x to get a third solution:
 

*Inverse of add 2 is sub. 2
 

*Inverse of mult. by -4 is div. by -4
 

 
 

Solutions:
  x y (x, y) 2 0 (2, 0) 0 -1 (0, -1) 1 -1/2 (1, -1/2)

Since the original problem has a <, this means it DOES NOT include the boundary line. 

So are we going to draw a solid or a dashed line for this problem? 
It looks like it will have to be a dashed line.

Putting it all together, we get the following boundary line for this problem:

 

 

An easy test point would be (0, 0).  Note that it is a point that is not on the boundary line. In fact, it is located above the boundary line. 

Let's put (0, 0) into the original problem and see what happens:
 

*True statement
 

 

Our solution lies above the boundary line.  This means we will shade in the part that is above it.

Note that the gray lines indicate where you would shade your final answer.

 

This is what we get when we intersect these two inequalities:

Note that the gray lines indicate where you would shade your final answer.

(return to problem 1d)
 

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Last revised on July 5, 2011 by Kim Seward.
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