### 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 ๐ฅ is greater than ๐. The second row would say ๐ฅ is less
than ๐. The third row is a slight variation
of the first row. This symbol tells us that ๐ฅ is
greater than or equal to ๐. And in a similar way, the fourth
row says ๐ฅ is less than or equal to ๐.

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,
weโll 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 ๐ฅ
plus one is greater than 10, then, to solve this inequality, we would need to find
all of the values that make ๐ฅ plus one greater than 10. And thatโs what weโre going to
learn how to do in this video.

Before we look at solving for
inequalities, letโs quickly remember the balance method when weโre working with
equations. An equation states the equality
between two expressions. We can represent it with a balance
scale. In this case, we have ๐ฅ plus five
equals eight. If we want to find just ๐ฅ, 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, weโll have a three and a five. Then weโre able to take that five
block away to see that ๐ฅ equals three. Algebraically, we represent that by
subtracting five from both sides of our equation.

Now, letโs look at how this applies
to inequalities. In this case, we want to say that
๐ฅ plus five is greater than eight. On the scale, that means the ๐ฅ
plus five side will weight more; it will be further down than the eight. In this case, when weโre solving
for ๐ฅ, weโll need to maintain this uneven scale. The ๐ฅ 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, weโll 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 ๐ฅ 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 ๐ฅ 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. Letโs look at some examples to see
how we apply this rule to simplify inequalities.

Given that ๐ฅ 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 ๐ฅ?

In order to solve for ๐ฅ, we want
๐ฅ to be the only thing on one side of the inequality. In this case, 34 is being added to
๐ฅ. If we wanted to get ๐ฅ 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 ๐ฅ must be
greater than or equal to 12.

Hereโs another example.

If ๐ฅ minus eight is greater than
negative three, then ๐ฅ is greater than blank.

We know that ๐ฅ minus eight is
greater than negative three. And we want to know then what value
must ๐ฅ be greater than. And to do that, weโll want to
simplify. Weโll want to get this ๐ฅ 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 ๐ฅ minus eight plus
eight just equals ๐ฅ. And that means we can add eight to
both sides of this inequality without changing its value. On the left, we have ๐ฅ and, on the
right, negative three plus eight equals five. And so we can say if ๐ฅ minus eight
is greater than negative three, then ๐ฅ must be greater than five.

Hereโs a third example.

If ๐ plus 46 is greater than or
equal to 39, then blank. (A) ๐ is greater than or equal to
seven. (B) ๐ is greater than or equal to
negative seven. (C) ๐ is greater than or equal to
85. (D) ๐ is greater than or equal to
negative 85. Or (E) ๐ is less than or equal to
negative seven.

Weโre starting out with the
inequality ๐ plus 46 is greater than or equal to 39. And we need to simplify it so that
we only have an ๐ 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 ๐, 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 ๐. ๐ plus 46 minus 46 equals ๐.

And here, we need to be really
careful. We have 39, but weโre subtracting
more than 39. Weโre 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 weโve brought down our
inequality symbol, because that remains true, to say that ๐ must be greater than or
equal to negative seven, which in this case is option (B).

In our next example, weโll have to
do a bit of rearranging.

Select the option that is
equivalent to ๐ plus seven is less than zero. (A) ๐ is greater than negative
seven. (B) ๐ is greater than or equal to
negative seven. (C) Negative seven is greater than
๐. Or (D) negative seven is greater
than or equal to ๐. (B) ๐ is greater than or equal to
negative seven. (C) Negative seven is greater than
๐. Or (D) negative seven is greater
than or equal to ๐.

First, we write down what we
know. ๐ plus seven is less than
zero. We can start to solve this problem
by isolating the ๐ 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, ๐ plus seven minus
seven equals ๐. And on the right, zero minus seven
equals negative seven. Weโve just found that ๐ is less
than negative seven. Now, at first glance, it doesnโt
look like we have ๐ is less than negative seven as an answer choice. To help us clear that up, letโs
consider this inequality on a number line.

Weโre dealing with negative seven,
and ๐ 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. Weโre saying that ๐ has to be
found to the left of negative seven on a number line. Using that information, letโs go
back and visit our answer choices. Option (A) says that ๐ is greater
than negative seven. And thatโs the opposite of whatโs
true. Option (B) says that ๐ is greater
than or equal to negative seven. Again, that is not true, since ๐
has to be less than negative seven. Option (C) is an inequality that
says negative seven is greater than ๐.

If we look at negative seven on our
number line, it is always true that negative seven is greater than ๐. Negative seven is greater than ๐
because ๐ is less than negative seven, which makes option (C) an equivalent
statement to ๐ is less than negative seven. When we look at option (D), it says
that negative seven is greater than or equal to ๐. But because the inequalities weโre
dealing with are only less than and greater than, they donโt have an equal-to
component, which makes option (D) false.

In our next example, weโre given a
simplified inequality, and weโll need to rearrange it.

Complete using less than or equal
to, less than, greater than, or greater than or equal to: If ๐ is less than or
equal to negative five, then ๐ plus one blank negative four.

We know that ๐ is less than or
equal to negative five. And we want to know what the
relationship of ๐ 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 weโll end up with ๐ plus one. We canโt simplify that any
further. And on the right, weโll end up with
negative five plus one. We have not changed the value of
our inequality. So we have ๐ 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 ๐ plus one
must be less than or equal to negative four.

In our final example, weโll 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, letโs 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 ๐ฅ
represent the money that Daniel will withdraw. And so weโre saying that ๐ฅ 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
๐ฅ 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. ๐ฅ must be greater than or equal to
65. And our ๐ฅ 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 weโve
written ๐ฅ plus 15 is greater than or equal to 80 and solved for ๐ฅ must then be
greater than or equal to 65.

Weโre now ready to summarize what
weโve 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 ๐ฅ plus three is greater
than six, we can subtract three from both sides of this inequality and say that ๐ฅ
is greater than three.