Video Transcript
In this video, we will learn how to
solve one-step linear inequalities using addition or subtraction. An inequality is a mathematical
sentence that contains one or more of the following inequality symbols. There are four inequality symbols,
and they show that the value of one expression is greater than the other. On the first row, we have a
greater-than symbol, which would say 𝑥 is greater than 𝑐. The second row would say 𝑥 is less
than 𝑐. The third row is a slight variation
of the first row. This symbol tells us that 𝑥 is
greater than or equal to 𝑐. And in a similar way, the fourth
row says 𝑥 is less than or equal to 𝑐.
Mathematical sentences that have an
equal sign are called equations. This equation says three plus two
is equal to five. In equations, both sides of the
equal sign have the same value. In an inequality, one side is
greater than the other. For example, here we have three
plus two is greater than four. When working with inequalities,
we’ll often need to solve them. And this means to find a range of
values that a variable can take to make the inequality true. If we look at the inequality 𝑥
plus one is greater than 10, then, to solve this inequality, we would need to find
all of the values that make 𝑥 plus one greater than 10. And that’s what we’re going to
learn how to do in this video.
Before we look at solving for
inequalities, let’s quickly remember the balance method when we’re working with
equations. An equation states the equality
between two expressions. We can represent it with a balance
scale. In this case, we have 𝑥 plus five
equals eight. If we want to find just 𝑥, we can
take the five block away. But in order to maintain a balance,
we would need to take five away from the other side as well. If we break up the original eight
that was on the scale, we’ll have a three and a five. Then we’re able to take that five
block away to see that 𝑥 equals three. Algebraically, we represent that by
subtracting five from both sides of our equation.
Now, let’s look at how this applies
to inequalities. In this case, we want to say that
𝑥 plus five is greater than eight. On the scale, that means the 𝑥
plus five side will weight more; it will be further down than the eight. In this case, when we’re solving
for 𝑥, we’ll need to maintain this uneven scale. The 𝑥 side should remain lower
than the eight side. If we try to remove something from
the heavier side, in order to keep the proportions the same, we’ll have to remove
the same amount from the lighter side.
And just like we did previously, we
can break that eight block into two pieces, a three and a five. And when we remove that five block,
we see a new inequality emerging. We see the statement that 𝑥 must
be greater than three. If we were to write this
algebraically, we would subtract five from both sides of the inequality, which leave
us with 𝑥 is greater than three. This balance method illustrates a
rule for inequalities. And the rule says this: An
inequality still holds true when the same number is added to or subtracted from both
of its sides. Let’s look at some examples to see
how we apply this rule to simplify inequalities.
Given that 𝑥 plus 34 is greater
than or equal to 46, what number do you need to add to both sides of the inequality
to solve for 𝑥?
In order to solve for 𝑥, we want
𝑥 to be the only thing on one side of the inequality. In this case, 34 is being added to
𝑥. If we wanted to get 𝑥 by itself,
we might subtract 34 from both sides of the inequality. And this is where we need to be
really careful reading this question because this question wants us to add something
to both sides of the equation. This first option, we have
subtracted 34. But if our question wants us to do
it in a format of addition, then we need to say we are adding negative 34. Since the question has asked what
number do we need to add to both sides of the inequality, the answer is we add
negative 34. And by adding negative
32 [34] to both sides of the inequality, we find that 𝑥 must be
greater than or equal to 12.
Here’s another example.
If 𝑥 minus eight is greater than
negative three, then 𝑥 is greater than blank.
We know that 𝑥 minus eight is
greater than negative three. And we want to know then what value
must 𝑥 be greater than. And to do that, we’ll want to
simplify. We’ll want to get this 𝑥 by itself
on the left side of the greater-than sign. We know that inequalities remain
true as long as we add or subtract the same amount from either side of the
inequality. I know that 𝑥 minus eight plus
eight just equals 𝑥. And that means we can add eight to
both sides of this inequality without changing its value. On the left, we have 𝑥 and, on the
right, negative three plus eight equals five. And so we can say if 𝑥 minus eight
is greater than negative three, then 𝑥 must be greater than five.
Here’s a third example.
If 𝑎 plus 46 is greater than or
equal to 39, then blank. (A) 𝑎 is greater than or equal to
seven. (B) 𝑎 is greater than or equal to
negative seven. (C) 𝑎 is greater than or equal to
85. (D) 𝑎 is greater than or equal to
negative 85. Or (E) 𝑎 is less than or equal to
negative seven.
We’re starting out with the
inequality 𝑎 plus 46 is greater than or equal to 39. And we need to simplify it so that
we only have an 𝑎 term to the left of our inequality symbol. And we know when working with
inequalities, as long as we add or subtract the same value from both sides, the
inequality remains true. If we want to isolate 𝑎, we can
subtract 46 from the left side of the inequality. But if we subtract 46 from the left
side of the inequality, we need to subtract 46 from the right side of the
inequality. On the left, that just leaves us
with 𝑎. 𝑎 plus 46 minus 46 equals 𝑎.
And here, we need to be really
careful. We have 39, but we’re subtracting
more than 39. We’re subtracting 46. Mathematically, that means we
subtract 39 from 46, which gives us seven. But then it takes the sign of the
negative 46. 39 take away 46 is negative
seven. And we’ve brought down our
inequality symbol, because that remains true, to say that 𝑎 must be greater than or
equal to negative seven, which in this case is option (B).
In our next example, we’ll have to
do a bit of rearranging.
Select the option that is
equivalent to 𝑎 plus seven is less than zero. (A) 𝑎 is greater than negative
seven. (B) 𝑎 is greater than or equal to
negative seven. (C) Negative seven is greater than
𝑎. Or (D) negative seven is greater
than or equal to 𝑎. (B) 𝑎 is greater than or equal to
negative seven. (C) Negative seven is greater than
𝑎. Or (D) negative seven is greater
than or equal to 𝑎.
First, we write down what we
know. 𝑎 plus seven is less than
zero. We can start to solve this problem
by isolating the 𝑎 variable. To do that, we subtract seven from
the left side of the inequality. And to maintain this true
inequality, that means we have to subtract seven from the right side of the
inequality. On the left, 𝑎 plus seven minus
seven equals 𝑎. And on the right, zero minus seven
equals negative seven. We’ve just found that 𝑎 is less
than negative seven. Now, at first glance, it doesn’t
look like we have 𝑎 is less than negative seven as an answer choice. To help us clear that up, let’s
consider this inequality on a number line.
We’re dealing with negative seven,
and 𝑎 has to be less than that. On a number line, less than
negative seven would be more negative values. And that would be an arrow pointing
to the left. We’re saying that 𝑎 has to be
found to the left of negative seven on a number line. Using that information, let’s go
back and visit our answer choices. Option (A) says that 𝑎 is greater
than negative seven. And that’s the opposite of what’s
true. Option (B) says that 𝑎 is greater
than or equal to negative seven. Again, that is not true, since 𝑎
has to be less than negative seven. Option (C) is an inequality that
says negative seven is greater than 𝑎.
If we look at negative seven on our
number line, it is always true that negative seven is greater than 𝑎. Negative seven is greater than 𝑎
because 𝑎 is less than negative seven, which makes option (C) an equivalent
statement to 𝑎 is less than negative seven. When we look at option (D), it says
that negative seven is greater than or equal to 𝑎. But because the inequalities we’re
dealing with are only less than and greater than, they don’t have an equal-to
component, which makes option (D) false.
In our next example, we’re given a
simplified inequality, and we’ll need to rearrange it.
Complete using less than or equal
to, less than, greater than, or greater than or equal to: If 𝑏 is less than or
equal to negative five, then 𝑏 plus one blank negative four.
We know that 𝑏 is less than or
equal to negative five. And we want to know what the
relationship of 𝑏 plus one would be to negative four. We know about inequalities that if
we add the same number to both sides of an inequality, it will still hold true. If we try to add one to both sides
of the inequality, on the left we’ll end up with 𝑏 plus one. We can’t simplify that any
further. And on the right, we’ll end up with
negative five plus one. We have not changed the value of
our inequality. So we have 𝑏 plus one is less than
or equal to negative five plus one. And we know that negative five plus
one is negative four. And so we confirm that 𝑏 plus one
must be less than or equal to negative four.
In our final example, we’ll need to
first write an inequality before we solve it.
Daniel wants to buy groceries. He wants to have at least 80
dollars cash with him. And he currently has 15
dollars. Write and solve an inequality that
would determine how much cash Daniel should withdraw from the bank.
Daniel wants at least 80 dollars
cash. To find out what kind of inequality
we should write, let’s think about this. Would it be okay if Daniel had more
than 80 dollars cash? Yes. Would it be okay if Daniel had
exactly 80 dollars cash? Yes. It would not be okay if he had less
than 80 dollars cash. And that means we need an
inequality symbol that represents more or equal. We need the money Daniel has to be
greater than or equal to 80 dollars. And the money Daniel has consists
of two pieces. The money he should withdraw and
the 15 dollars he already has. Together, these two values should
be greater than or equal to 80 dollars. We can let the variable 𝑥
represent the money that Daniel will withdraw. And so we’re saying that 𝑥 plus 15
must be greater than or equal to 80.
At this point, we have successfully
written an inequality to show the money that Daniel needs. But we need to continue by solving
this inequality. We want to solve for what values of
𝑥 would this statement be true. And we do that by subtracting 15
dollars from both sides of the inequality. 80 dollars minus 15 dollars equals
65 dollars. 𝑥 must be greater than or equal to
65. And our 𝑥 represents the money
that he should withdraw from the bank. Daniel must withdraw at least 65
dollars if he wants to have 80 dollars cash with him. And so an inequality that we’ve
written 𝑥 plus 15 is greater than or equal to 80 and solved for 𝑥 must then be
greater than or equal to 65.
We’re now ready to summarize what
we’ve learned. An inequality is a mathematical
sentence containing an inequality symbol. It shows that the value of one
expression is greater than that of the other. To simplify inequalities, we can
use the addition and subtraction rule for inequalities, which tells us that an
inequality still holds true when the same number is added to or subtracted from both
of its sides. Just note that it says equality
here, where it should say inequality. An inequality still holds true when
the same number is added to or subtracted from both of its sides. And so, if 𝑥 plus three is greater
than six, we can subtract three from both sides of this inequality and say that 𝑥
is greater than three.
