Solve the inequality seven 𝑦 plus
three is greater than two 𝑦 minus four.
In this question, we’re being asked
to solve an inequality rather than an equation. It’s a linear inequality because
the highest power of the variable which in this question is 𝑦 is just one. When we solve an inequality, we get
a range of possible values for 𝑦 as our answer rather than just one value which is
what we’d get if we solved a linear equation.
The steps involved in solving a
linear inequality are very similar to the steps involved in solving a linear
equation. First, we noticed that there are 𝑦
terms on both sides of the inequality. So our first step is going to be to
group them on the same side. To do this, we want to group on the
side which has the larger number of 𝑦 to start off with so that we end up with a
positive number of 𝑦. So we’re going to group on the
In order to do so, we need to
subtract two 𝑦 from each side as this will cancel out the positive two 𝑦 on the
right-hand side. Seven 𝑦 minus two 𝑦 is five
𝑦. We bring down the plus three and
the inequality sign. On the right, two 𝑦 minus two 𝑦
is just zero and we bring down the negative four. So we now have five 𝑦 plus three
is greater than negative four.
The next step is to subtract three
from each side because we’re trying to leave 𝑦 on its own on the left-hand
side. And by subtracting three, this will
cancel out the positive three that we have here. Five 𝑦 plus three minus three is
just five 𝑦. And on the right-hand side,
negative four minus three is negative seven. We’re taking away three. So we’re moving further into the
The final step in solving for 𝑦 is
to divide both sides of this inequality by five as this will eliminate the factor of
five on the left-hand side. Five 𝑦 divided by five is just
𝑦. And on the right-hand side, we can
leave our answer as a fraction: negative seven over five. So we found a range of values for
𝑦 as our answer. 𝑦 can take any value as long as
it’s greater — that’s strictly greater — than negative seven over five.
If you want, you could also convert
this improper fraction into a mixed number, in which case, it would be negative one
and two-fifths. So our answer would be 𝑦 is
greater than negative one and two-fifths. Or you could convert two-fifths to
a decimal. It’s equal to 0.4, in which case,
our solution would be 𝑦 is greater than negative 1.4. As we haven’t been asked to give
our answer in a particular form, any of these three formats would be acceptable.
Now I said that solving a linear
inequality was a lot like solving a linear equation and that’s true. But there is one key difference
which we need to be aware of although it wasn’t relevant here. What we need to remember is that if
we multiply or divide both sides of an inequality by a negative number, we must make
sure that we reverse the direction of the inequality. So if it was a greater than sign,
we change it to a less than sign and vice versa.
To see why this is true, let’s take
the simple inequality zero is less than seven which is certainly true. If we multiply the numbers on each
side of this inequality by negative one, well zero multiplied by negative one is
still zero and seven multiplied by negative one is negative seven. But if we don’t reverse the
direction of the inequality, then it now says zero is less than negative seven. This is clearly false. So in order to make the statement
true, we have to reverse the direction of the inequality sign and change it to say
zero is greater than negative seven.
You must make sure that you do this
whenever you multiply or divide any inequality by a negative number. All the other steps in solving a
linear inequality are the same as they are in solving a linear equation. Our solution to this inequality is
𝑦 is greater than negative seven over five.