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## Algebra 1

### Course: Algebra 1 > Unit 2

Lesson 5: Multi-step inequalities# Using inequalities to solve problems

We can use inequalities to solve problems in a given context. Created by Sal Khan.

## Want to join the conversation?

- How many stops can Kalya make before spending $14.50 if 1 stop is $1.50(9 votes)
- Well, when you think about it... She can stop 14 times... (or 13 im too tired to think) because everytime she stops she is loosing another $1.50.

1- 1.50

2- 3

3- 4.5

4- 6

5- 7.5

6- 9

7- 10.5

8- 12

9- 13.5

10- 15 (which stops, cause you can't go over $14.50 dollars.

So now that I used my brain... It would be 9 times... xD*I hope this helps! <3*(20 votes)

- How would you graph this inequality? 🤔(7 votes)
- You put a closed circle and you draw an arrow pointing to the left.(6 votes)

- Why weren't the inequalities swapped when you divided??(3 votes)
- You see we only swap when we divide by a negative. Hope this helps.(14 votes)

- the 8 kilometers has nothing to do with it right?(3 votes)
- Yes, it is just explanation of the problem(7 votes)

- My teacher said you flip the inequality sign or something whenever you move a number to another side. Is this true?(1 vote)
- I do believe it is only when you divide or multiply by a negative number. 😃(8 votes)

- At3:26, why does Sal round down?(2 votes)
- Because the problem requires you to find the largest number of stops Kayla can buy, without spending more than 15$. And you did find that number of stops: 7.6 stops,
**BUT**you cannot buy 7.6 stops because the answer being in decimals goes against the policy of the supposed station here, which is to pay**1.25$ per stop**! You would only pay 0.75$ ([7.6*1.25]-[7*1.25]) and not the full**1.25$ per stop**! And rounding up to 8 stops would mean spending 15.5$ (8*1.25+5.5) which is**MORE**than the 15$ Kayla wants to spend. So rounding down to 7 stops is the only solution left, that fulfills the condition of this problem.(7 votes)

- How do I do this? Sal usually makes it pretty clear to me what to do, but right now I am very confused. Can someone explain how to do this in different words?(0 votes)
- First, you have to learn the language of inequalities:

> (more than, greater than, larger than, above)

< (less than, smaller than, below)

≥ (more than or equal, at least, no less than, minimun)

≤ (less than or equal, at most, no more than, maximum)

In this problem the language is "doesn't (want to spend) more than" which indicates not more than or ≤. The rest is very similar to a equality, y=mx+b, or more appropriate in this situation mx+b=y, the 5.50 is a set fee (b) and the slope is 1.25 per stop (s). Putting these in and changing the = to a ≤ gives the equation 3.3+1.25s≤15.

This might not quite answer your question, so is there any follow up question?(6 votes)

- How do you graph the inequality?(1 vote)
- You put a CLOSED circle (it means that 7.6 is included) and you draw an arrow pointing to the left.

PS: This is for the number line(3 votes)

- How do you know how which way the greater than or less than sign or what side it should be turned to based on the given information.(1 vote)
- If you mean knowing which way the inequality points after reading the problem, you need to be aware of keywords, like "at most" or "at least", and do some mathematical reasoning to figure out the rest. Knowing the directions of the inequalities also comes with experience--just doing a bunch of these types of problems. Be patient..(1 vote)

- how do i know what sign to put when they dont really say more than or less than(1 vote)
- Not a really practical method that I am mentioning here but you would need to practice a ton to get the hang of it. Mathematical reasoning as they say. Find the keywords for each particular sign.

A list made by another user (a "David Severin") might help:

> (more than, greater than, larger than, above)

< (less than, smaller than, below)

≥ (more than or equal, at least, no less than, minimun)

≤ (less than or equal, at most, no more than, maximum)(1 vote)

## Video transcript

- [Instructor] We're told that
Kayla wants to visit a friend who lives eight kilometers away. She'll ride the subway as far as she can before walking the rest of the way. First, she needs to buy an
access pass that costs $5.50. There's also a fee of $1.25 per stop. This is an expensive subway. Kayla doesn't want to spend
more than $15 on the trip. So she wants to know the largest number of stops she can afford. Let S represent the number
of stops that Kayla buys. So first, pause this video and see if you can write an inequality that describes how many stops, or that describes the
situation that describes that she wants to take
as many stops as she can, but she doesn't wanna spend more than $15. All right, now let's do this together. So first, let's just
think about an expression for how much she spends. So no matter what, she's
going to spend $5.50, so we can write it like this, so $5.50, that's what she's going to spend, even if she doesn't take any stops. And it's $1.25 per stop, and
S is the number of stops. So the amount she's going
to spend just from the stops is going to be $1.25 times S. So it's going to be plus $1.25 S. This is the upfront she has to spend, and this is how much she's
going to spend on stop. So this is an S right over here and I wrote a five right next to it, they look kind of similar. And we know that she doesn't
want to spend more than $15. So she's willing to spend up to $15. So this total amount that she spends has to be less than or equal to $15. Or if we didn't write it
with the dollar symbols, we would write 5.50 plus 1.25 S is going to be less than or equal to, or needs to be less than or equal to 15. Now that we've written this inequality, what is the number of stops
that Kayla can afford? What's the largest number of
stops that she can afford? Pause this video and
try to figure that out. Well, to figure that out,
we just have to solve for S and then figure out what the largest S is that satisfies the inequality
once we've solved for S. So the first thing I would do is subtract 5.50 from both sides. When we do that, we are
left with 1.25 or $1.25 S is less than or equal to 9.50. And then I would divide
both sides by 1.25. And since I'm dividing both
sides by a positive value, it doesn't change the
direction of the inequality, 1.25 and then divide this by 1.25, 9.5 divide by 1.25 is equal to 7.6. So we get that S needs to be
less than or equal to 7.6. So we can't take a
fractional number of stops. So the largest number of
stops that Kayla can take is going to be seven stops. She can't take eight, and she can't take 7 1/2 or 7.6. So the largest number she
can take is seven stops. So she can take as many as seven stops. And we are done.