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## Algebra (all content)

### Unit 2: Lesson 14

One-step inequalities# One-step inequality involving addition

How to solve and graph one-step inequalities. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- What if both sides of the inequality are negative? Does the sign switch?(44 votes)
- This is not a multiplication or division so it does not switch.(4 votes)

When would would need to flip the direction of the inequality(34 votes)- When you divide or multiply both sides of the inequality by a negative number. For example: -4x > 9 Here you have to divide both sides by a negative number, negative 4, so you carry out the division just like you would in a regular equality, but the only thing you do differently is you flip the inequality sign. So in this case it would be: x < -9/4. Hope this helps!(46 votes)

- When do you need to flip the inequality?(6 votes)
- You are multiplying or dividing by a negative. For example: the equation 7>1 does not hold when you multiply both sides by negative one, as that would leave you with -7>-1, which is FALSE. Take any two numbers and multiply or divide them by a negative, and see what happens:)

Hope that helped! :D(10 votes)

- how do you know when the signs switch?(1 vote)
- Dylan,

You only need to switch the inequality sign when you multiply or divide both sides of the inequality by a NEGATIVE number. I find this sometimes makes more sense if we take all of the variables away and just look at numbers.

For example, 3 < 4, right? Well what happens if I multiply both sides by -2, and I DON'T remember to switch the signs? I get -6 < -8โฆ but -6 ISN'T less than -8. -8 is Less than -6. So, since I multiplied by a negative number, if I switch the signs the statement becomes true again: -6 > -8.

Suppose I divide by a negative number? The same thing happens. Watch:

Let's say I start with 12 > 9. Clearly that is true, 12 IS greater than 9. But suppose I divide both sides by -3 and forget to switch the signs? Then I get:

-4 > -3. Well that's not true anymore. -4 ISN'T greater than -3, it's LESS than -3. That's why I have to switch the sign: -4 < -3.

You'll notice that this only happens when I multiply or divide by a NEGATIVE number. If I multiply or divide by a positive number, I don't need to do anything: 1 < 2 is a true statement. 1 IS less than 2. And if I multiply both sides by POSITIVE 4, I get 4 < 8, which is STILL a true statement.

I also don't need to switch the signs if I Add or subtract ANY numbers, positive or negative: 6 < 9 is true. And if I subtract 5 from both sides (or add -5) I get: 1 < 4. Still true, and I didn't need to switch the signs.

It can be harder to see this when there are a bunch of variables in the expression. But if you ever forget, just take the variables out and use numbers, you can check to see when you need to switch the inequality sign all by yourself just like we did above. It shouldn't take you long to prove to yourself that you only switch the signs when you multiply or divide by a negative number.

Does that help, Dylan?(11 votes)

- Can someone help here? When do you flip the sign? I never get these correct!(3 votes)
- When you divide by a negative number the sign flips.(3 votes)

- Why do you have to flip the sign when you divide by a negative?(2 votes)
- Multiplying or dividing by a negative is like multiplying -1 and then multiplying or dividing a positive normally. Thus, I'll just explain why multiplying by -1 makes you flip the sign to explain all of it.

When you multiply by -1, you are flipping the numbers on both sides of the inequality over 0. Remember that -1 is greater than -2 because -1 is**less**left of 0 on the number line. If you take the negative of each side of the inequality, you have to flip the sign because lesser positive numbers before will become greater than greater positive numbers before. When you flip it over 0, you're switching the numbers places so that one number is left of the other [on the number line] when it was right of it [on the number line] before and vice versa. Let's do some examples to show you this.

A number with less absolute value will be less when positive, but more when negative.

1<2-->-1>-2

Positive numbers are always greater than negative numbers, just as negative numbers are always less than positive numbers

2>-1-->-2<1

A number with greater absolute value will be less when negative, but greater when positive.

-2<-1-->2>1

I hope this helps!(3 votes)

- What if I want to write the solution set for this question?(2 votes)
- In the same format as the previous videos, I believe the solution set would be something like this:
`{ x is a real number | x โค -2 }`

which could be read as`"x is a real number such that x is less than or equal to negative two."`

(3 votes)

- Can I also express the final answer xโค-2 as (negative infinity, -2] in interval notation?(1 vote)
- Yes, you can use -infinity in interval notation.(2 votes)

- can you explain compound inequalities more(2 votes)
- more on multi step inequalities(2 votes)

## Video transcript

Solve for x plus 8 is
less than or equal to 6 and graph the solution. So we have x plus 8 is
less than or equal to 6. So let's solve this
inequality for x. And the easiest way to isolate
an x on-- let's isolate on the left-hand side
since it's already there-- is to just
get rid of this 8. And the best way to get
rid of this positive 8 is to subtract 8 from both
sides So let's subtract 8 from both sides. That won't change the
direction of the inequality. So the left-hand side--
x plus 8 minus 8. And you're just left with an x. The right-hand side-- 6
minus 8 is negative 2. And we still have the
less than or equal. So we solved the inequality. We have x is less than
or equal to negative 2. So let's draw that
on a number line. So that's my number line. Let's stick 0 over here. Maybe if we go 1,
and then we could go negative 1,
negative 2, negative 3. And we could keep
going to the left. And we want all of
the x's that are less than or equal
to negative 2. Since it can be
equal to negative 2, we'll put a filled-in line
right here at negative 2, and all of the values
less than that. If it was just less than, if
there wasn't the equal sign, we would have an open dot. But since this is
less than or equal to, we've closed this dot. And then we want all of
the values below that. And you could just sample a
few and verify for yourself that they work. Based on this,
negative 3 should work. And if you took negative
3-- negative 3 plus 8 is 5, which is definitely
less than 6, so that works. And negative 1 shouldn't work. It's not included in
this set over here. So let's try that out. Negative 1 plus 8 is 7, which
is definitely not less than 6. So just sampling
some points, it seems like we've got the
right solution.