Graphing Linear Inequalities 05:29 minutes

Video Transcript

Transcript Graphing Linear Inequalities

Lately, but possibly everyday, My Dear Aunt Sally has been so forgetful. For my birthday, she baked 15 cookies, trimming them with edible gold leaf, and she squeezed a bowl of lemons to make 10 glasses of lemonade, but then my mom reminded her that my birthday is not till next month.

Rather than toss the tasty treats, she decides to sell them. At the very least, she wants to recover the 50 dollars she spent on ingredients. So, how much should she charge for each treat? Let's help her.

Write a Linear Inequality in Slope-Intercept Form

Prior to evaluating My Dear Aunt Sally’s problem, let's write a linear inequality in slope-intercept form, then graph the inequality to determine the solution set.

What are the facts? Aunt Sally wants to sell 15 cookies and 10 glasses of lemonade. She needs to collect at least 50 dollars, so we should write the inequality using the greater than or equal to symbol.

The sum of 15 cookies for an unknown price, x, plus 10 glasses of lemonade for an unknown price, y, has to be greater than or equal to 50 dollars.

To put this in slope-intercept form, we have to move the terms around. First, using the opposite operation, move 15x to the other side of the inequality. Next, isolate the y by dividing both sides by ten.

Do we need to flip the inequality sign? No, since we're dividing by a positive number, we’re ok. Always remember to check the sign because if you multiply or divide by a negative number, you'll need to flip it.

Okay, where are we? Right, y ≥ −1.5x + 5. We're good to go!

Graphing a Linear Inequality in Slope-Intercept Form

Now that the inequality is in slope-intercept form, let’s graph it! The y-intercept is 5, and the slope is negative one point five. We know how to draw the line to indicate equal to but how do we represent greater than?

All the ordered pairs above the line will result in an answer that is greater than the equation we set up. Let’s shade this area.

To be certain this is correct, let’s check our work. Pick a point on the graph, any point. How about the ordered pair 1, 3? This point is below the line.

Will Aunt Sally collect 50 dollars or more if she sells 15 cookies for 1 dollar each and 10 glasses of lemonade for 3 dollars each? Do the math.

The product of 1 and 15 plus the product of 3 and 10 is equal to 45. No, this price plan won’t work and no wonder, the point 1, 3 is not in the area of values we shaded!

Any point along the line or in the shaded area will give Aunt Sally the pricing combo she needs to earn fifty dollars or more – that is, if she sells all of the tasty treats.

More Examples Graphs of Inequalities in Slope-Intercept Form

Now that you understand the concept, let’s look at some other inequalities and their graphs.

  • For y is less than or equal to x plus 2, notice the shaded area is below the solid line. All values on and below this line are in the solution set.
  • For y is greater than negative one-half x plus five, we use a dashed line as, the values on the line are not part of the solution set. Only the values above the dotted line solve the inequality.
  • For y is less than negative x plus 4, we have to use a dashed line; but this time, the values below the dashed line are in the solution set.

Let’s make a graphic organizer to help you remember.

  • For y is greater than or equal to, the line is solid, and the values on and above the line satisfy the inequality.
  • For y is less than or equal to, the line is solid, and the values on and below the line are part of the solution set.
  • For y is greater than, we must use a dashed line, and only the values above the line are part of the solution set and lastly, for y is less than, the line is dashed and only the values below the line satisfy the inequality.

Summary: Steps to Solve Linear Inequalities by Graphing

Let’s summarize the steps to solve inequalities:

  • Write your equation in slope-intercept form.
  • While isolating the variable, if needed, flip the inequality symbol.
  • Draw the appropriate line, solid or dashed.
  • Determine if the solution set includes the area above or below the line, then pick any point and check your work.

Predictably excited, My Dear Aunt Sally sold all of the cookies and all the glasses of lemonade! She’s ecstatic, but wait. You won’t believe this! It’s Aunt Sally’s birthday! How did she forget her own birthday?!