In this lesson, what we’re looking
to do is solve multistep inequalities. And these are exactly what they
sound like, inequalities that take multiple steps to solve. And when we deal with inequalities,
we have a set of notation. And I’ve shown here with our pink
arrow things like less than, greater than, less than or equal to, or greater than or
equal to. So we’ll have a little look at how
we’d use that.
Well, with something like this, you
might think, “Well, where do I begin? What would I do if I want to solve
a multistep inequality?” Well, in fact, it’s very similar to
solving equations. We follow a lot of the same
steps. We just deal, like I said, with
this notation, which is our inequality notation.
Well, the first thing we want to
have a look at is the notations involved with our inequalities. So first of all, we’ve got 𝑥 is
less than 𝑦. And we can see that because the
pointy end is pointed towards the 𝑥 and the wide-open end is pointing towards the
𝑦. And we always have the open side
pointing towards the larger number. Then the next one down is 𝑥 is
greater than 𝑦 cause we can see that the inequality sign has been flipped the other
Then what we have is 𝑥 is less
than or equal to 𝑦. And this is really key. So we’ve got 𝑥 is less than or
equal to 𝑦. The bit that says that it’s “or
equal to” is the little line at the bottom of our inequality sign. So this means that 𝑥 could be
anything that’s less than 𝑦 or it could be 𝑦 itself. And then, finally, we have 𝑥 is
greater than or equal to 𝑦. So great, we’ll now take a look at
the notation we’re going to use. So let’s get on and have an example
and see if we can solve a multistep inequality.
Find the solution set of the
inequality negative 14𝑥 minus 52 is less than or equal to negative 18𝑥 in the set
of real numbers. Give your answer in interval
Well, the first strange thing we
want to see in this question is this R-looking character. And what this means is all real
numbers. So we’ll know that the results of
our inequality is gonna be a real number. Well, you might think, “Well,
aren’t all numbers real?”
Well, in fact, no, there are
imaginary numbers that we deal with in maths. But we deal with them later on in
the maths course. So now when we’re gonna solve our
inequality, we’ll do it in the same way that we’d solve an equation. So we’ve got negative 14𝑥 minus 52
is less than or equal to negative 18𝑥. So what we’re gonna look to do now
is add 18𝑥 to each side of the inequality and also add 52. And when we do that, what we’re
gonna get is four 𝑥 is less than or equal to 52.
And you can see here I’ve completed
two steps. Well, this is in fact a multistep
inequality because we’ve done two steps. But we haven’t quite finished yet
because we’ve got another step that we need to do to solve our inequality.
So the final step is to divide
through by four. And that’s because we want to find
out what one 𝑥 is. Well, when we do that, we’re gonna
get 𝑥 is less than or equal to 13. So great, we’ve solved the
inequality. So have we solved the problem?
Well, no, not quite, because the
question asked us to give our answer in interval notation. So what will this be, in interval
notation? Well, when we write this in
interval notation, we have this here. We’ve got parentheses, then
negative ∞ comma 13, and then a square bracket. So let’s think what all of this
Well, the first of all, on the
left-hand side, we have a parenthesis. And that’s because we know that if
our 𝑥 is less than or equal to 13, then it can go all the way down to negative
∞. But it would not include negative
∞. Then, we can see on the right-hand
side, we’ve got a square bracket. And this means that it does include
13 because we were told that 𝑥 is less than or equal to 13. So we know that our values can take
any value from negative ∞, but not including negative ∞, all the way up to 13, but
also including 13.
So this first question has helped
us with some of our skills because we’ve already looked at solving multistep
inequalities. So we’ve done that. We’ve also talked about some set
notation as we’ve talked about ℝ, meaning all real numbers. And then we’ve given our answer in
interval notation. So we’ve covered a lot of
things. But let’s see where we can go to
Well, what we’ll look at next is a
bit more set notation. Also, we’re gonna look at
distributing across parentheses in our inequality. And also then we’ll look at leaving
our inequality or the answer to it in a different format.
Solve the inequality 10𝑥 plus 16
is less than or equal to eight multiplied by 𝑥 minus 19 in the set of rational
And as we said when reading out the
question, this ℚ means rational numbers. And what we mean by rational
numbers are any numbers that can be represented using a fraction. Okay, so now let’s solve our
Well, the first step or stage in
our multistep inequality is to distribute across our parentheses. And what this means is we’re gonna
multiply the eight by both terms inside our parentheses. So we’re gonna get 10𝑥 plus 16 is
less than or equal to. Then we’ve got eight 𝑥 cause eight
multiplied by 𝑥 is eight 𝑥. And then we’re gonna get minus
152. And that’s cause eight multiplied
by negative 19 gives this negative 152.
So now what we can do to make sure
that we have the 𝑥s on one side of our inequality and numerical values on the other
side of our inequality is to subtract eight 𝑥 and 16 from each side. So when I do this, what I’m gonna
get is two 𝑥 is less than or equal to negative 168. And then all we need to do is
divide both sides of the inequality by two. And we get 𝑥 is less than or equal
to negative 84.
So what this means is, 𝑥 is any
value that’s less than negative 84 but also including negative 84. Well, we’ve just shown the answer
using inequality notation cause 𝑥 is less than or equal to negative 84. We could also use some interval
notation as I’ve shown here, where we’ve got a parenthesis. Then we’ve got negative ∞, then
comma negative 84. And then we’ve got a square
bracket. And what this means is, the values
can take any value from negative ∞, but not including negative ∞, all the way up to
negative 84, and including negative 84.
And then we also have this other
way of representing our answer that tells us that 𝑥 is an element or a member of
the rational numbers such that 𝑥 is less than or equal to negative 84.
So again, we built on our skills
because what we’ve done is shown some interval notation. And we’ve also started to
distribute across parentheses.
So now for the final type of this
example, what we’re gonna do is look at an inequality that has parentheses on both
side. So we’re gonna have to distribute
across the parentheses. We’re gonna have to simplify. And again, we’re gonna give our
answer in this form, which is involving some interval notation.
Solve the inequality nine 𝑥 minus
three multiplied by negative seven 𝑥 plus nine is less than negative seven
multiplied by negative nine plus 𝑥 minus two in the set of rational numbers.
So now as you might remember from
set notation, this ℚ means rational numbers. Rational numbers are numbers that
can be represented by a fraction.
Now to solve this problem, what we
need to do, first of all, is distribute across our parentheses on both sides of our
inequality. And that’s because we’re gonna
solve it in the same way we’d solve an equation. So on the left-hand side, we’re
gonna get nine 𝑥 plus, and then we’ve got 21𝑥. This is positive 21𝑥 because we’ve
got negative three multiplied by negative seven 𝑥. And then we have minus 27. And this gonna be less than 63. And that’s because negative seven
multiplied by negative nine gives us 63 cause a negative multiplied by a negative
gives us a positive. And then we’ve got minus seven 𝑥
Okay great, so now what we need to
do is simplify both sides. So when we do that, we’re gonna get
30𝑥 minus 27 is less than 61 minus seven 𝑥. So now the next step is to get the
𝑥s all on one side of the inequality and numerical values on the other. So what I’m gonna do to do that is
add seven 𝑥 to each side of the inequality and add 27 to each side of the
inequality. So when we do this, we’re gonna get
37𝑥 is less than 88.
So finally, all that’s left to do
is divide each side of the inequality by 37. When we do that, we get 𝑥 is less
than 88 over 37. And then we can show that using our
interval notation such a way that 𝑥 is a member of the set of rational numbers such
that 𝑥 is less than 88 over 37.
So now we’ve covered a problem
that’s got parentheses on both sides. So we’ve dealt with a number of
different multistep inequalities. But what’s the next step? Well, now is the point where we
might think, “Well, this is great, but what’s the point of solving
So as we’ve said, what are
inequalities for? How can they be used in real
life? Well, what inequalities are
actually used for is a variety of things, and particularly in things like business
where it can be help with stock control or wages or engineering when we’re looking
at tolerances for instance, the number of people that can fit in a lift or the
tolerances involved in a bridge. And we do that using something
called linear programming, which is another method where we can use inequalities to
help us solve problems. And inequalities can also be
something that we graph for as well.
Well, for our final example, let’s
take a look at a question that looks at solving inequalities in context. So we’re gonna have a look at a
A cell phone company offers the
following two plans. Plan A, which is 15 dollars per
month and two dollars for every 300 texts. Plan B, 25 dollars per month and
0.50 dollars for every 100 texts. How many texts would you need to
send per month for plan B to save you money?
Well, the way that we can set up
and solve this problem is to first think, how much does it cost for plan A and plan
B to send 100 texts? Well, if we take a look at plan A,
it costs two-thirds of a dollar per 100 texts. And that’s because of its two
dollars for every 300 texts. Just divide that by three. Then if we look at plan B, we can
see that it’s half a dollar or 50 cents for every 100 texts.
Well, what we can do is we start
set up an inequality. And we could do that because what
we’re gonna have for plan A is 15 plus two-thirds 𝑥, where 𝑥 is the number of
hundreds of texts. Well, we want to see where this is
greater than. And we want to see where it’s
greater than because what we’re trying to do is find out how many texts you’d need
to send per month for plan B to save you money. So we want plan B to be less
than. And then we’ve got for plan B 25
plus a half 𝑥.
So now what we do is we’ve
converted both of our fractions into sixths to make it have a common
denominator. Now we’ve got 15 plus four-sixths
𝑥 is greater than 25 plus three-sixths 𝑥. So now what we’re gonna do is
subtract 15 and subtract three-sixths 𝑥 from both sides of the inequality. And we’re gonna do this because we
want the 𝑥 on one side. And then we want the numeric values
on the other. And when we do that, we’re gonna
get a sixth 𝑥 is greater than 10.
So if we’ve got a sixth 𝑥 is
greater than 10, if we multiply both sides by six, we’re gonna get 𝑥 is greater
than 60. Well, as we said at the beginning,
𝑥 represents the number of hundreds of texts. So therefore, the number of texts
that would need to be sent for plan B to save you money would be greater than 6000
texts. And we get that because we
multiplied 60 by 100, which gives us 6000.
So now the final skill that we’re
looking at was how to solve a multistep inequality in a real-life context. And we’ve done that. So we’ve reached the end of what we
were trying to look at. So now let’s have a recap of
everything that we’ve learned in this lesson.
So the key points from this lesson
are, first of all, a multistep inequality is one that requires more than one step or
stage of algebraic manipulation to solve. Secondly, inequalities are solved
using a similar balancing method to equations. So if we see an inequality, don’t
get scared and think, “What am I gonna do?” Because all you do is you solve it
in the same way as you solve an equation. Just bearing in mind that you have
an inequality sign instead.
Next, we have our notation, less
than, greater than, less than or equal to, or greater than or equal to. And these are all signs that we use
in inequalities. It’s worth remembering that the
difference between them is the line underneath, the-the line at the bottom. And this tells us that it can be
greater than or less than or equal to. So this is it can be or equal
It’s also worth remembering the
bigger sides or the open side of our inequality sign is to the bigger number. And the pointy end is to the
smaller number. Well then, what I’ve shown is a
little bit of our interval notation. That’s our parentheses and our
brackets. If we’ve got a parenthesis, this
means that it doesn’t include the value next to it. So this is like our greater than or
less than. However, if we’ve got a square
bracket, this means that it does include that value. So this is always equal to.
Next, we know that inequalities can
be used in real-life contexts. So we’ve shown that with our last
example. And finally, inequalities can be
represented using different notation. And I’ve talked a little bit about
that with our interval notation. We’ve got set notation and we’ve
also got our inequality notation itself.