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Solving and Graphing Inequalities using Multiplication or Division - Problem 2

Teacher/Instructor Alissa Fong
Alissa Fong

MA, Stanford University
Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

As is the case when solving equations, when x is in the numerator, this means that x is being divided by the value in the denominator. To undo what is being done to x, do the opposite -- multiply by the value. Once the variable is isolated, graph the inequality on a number line. When graphing, keep in mind when to use an open circle versus a closed circle, and which direction the arrow should be pointing.

Remember that when you are working with inequalities and you're solving them, it's almost exactly like solving equations except for sometimes you need to change the direction of the inequality.
Let's check this one out.

I have x divided by 4 is less than or equal to -8. Be really careful that when you see this fraction problem you don't just skip it, come on guys you can totally do this. If you want to you could treat this as 1/4 times x, that's the same thing or the way I think of it is as x divided by 4. The opposite of dividing by 4 would be to multiply both sides by 4 and so now I'm left with x is less than or equal to -32.

Now there's a negative involved in this problem so I want to just double check and see if I need to change the direction, let's see. You would change the sign if you multiplied or divided by a negative value but I didn't. I multiplied by a +4. So this sign stays the same. This is my final answer x is less than or equal -32.

Let's go ahead and draw it on the number line, -32 I'm going to put a closed circle because that's a less than or equal to, and then I want to mark all x values that are smaller than -32. Smaller numbers are out there to the left that means I could use -50, -100, -32.5 or even -32. Any one of those values would work when I substitute it into this original inequality in the x place.

So again guys, this is kind of a trick problem because there is some multiplying and dividing going on and there is a negative number but it's not the number I'm solving with. It's not the number I'm dividing or multiplying by. Just be really careful that you only change the sign if you multiply or divide by a negative value.

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