### Video Transcript

Which inequality has been graphed
in the given figure?

When we have an inequality
represented on a graph, we will always have a line, whether thatβs a complete line
or a dotted line, along with a shaded region. The first thing we will need to do
is identify the equation of the straight line. We can remember that the equation
of a straight line can be given as π¦ equals ππ₯ plus π, where π represents the
slope, or gradient, and π is the π¦-intercept. The π¦-intercept is usually very
easily identified from the graph. Itβs the point where the line
crosses the π¦-axis. On this graph, this happens at the
coordinates zero, negative three. And so the π¦-intercept is equal to
negative three. The slope of a line can be found by
the rise over the run. And we can work this out using two
coordinates π₯ one, π¦ one and π₯ two, π¦ two by the slope is equal to π¦ two minus
π¦ one over π₯ two minus π₯ one.

To make this process easier for
ourselves, we should select coordinates which are easily identifiable. Usually, these will have integer
values. We can see that the coordinates
zero, negative three lie on the line but so do the coordinates one, one. It doesnβt matter which coordinates
we designate with the π₯ one, π¦ one or π₯ two, π¦ two values. So letβs take π₯ one, π¦ one to be
the coordinates zero, negative three. The slope will therefore be equal
to one minus negative three over one minus zero. This simplifies to four over one,
which of course is equal to four. Since we know that the slope π is
four and the π¦-intercept is negative three, we have the equation of the line as π¦
equals four π₯ minus three.

However, this isnβt the final
answer since we were asked to write the inequality represented by the shaded
region. Instead of the equals sign, we will
need one of the inequalities, greater than, greater than or equal to, less than, or
less than or equal to. So how do we know which one of
these it will be? Well, we can notice that the line
that we were given is a dotted line. This means it is a strict
inequality. It will either be greater than or
less than. The two other inequalities that
involve equals to will not be a possibility.

We can now test which inequality it
will be by selecting a coordinate in the shaded region, which does not lie on the
line. Letβs pick the coordinate three,
two. At these coordinates, the π₯-value
is three and the π¦-value is two. We will therefore need to compare
the value of π¦, which is two, with the value of four π₯ minus three, which is four
times three minus three. When we simplify the right-hand
side, we get nine. We, of course, know that two is
less than nine. And therefore, the missing
inequality must be less than. The answer is therefore that the
inequality which has been graphed is π¦ is less than four π₯ minus three.

Itβs worth pointing out that
everything which is not in the shaded region and not on the line represents the
inequality π¦ is greater than four π₯ minus three. If we checked this using a
coordinate, for example, the coordinate zero, zero, plugging this into the
inequality we would have π¦ is equal to zero and π₯ is equal to zero. On the left-hand side of the
inequality, we would have zero. And on the right-hand side, we
would have negative three. Since this region is π¦ is greater
than four π₯ minus three, then the region which is shaded in is π¦ is less than four
π₯ minus three.