Question Video: Determining the Inequality Represented by a Given Graph | Nagwa Question Video: Determining the Inequality Represented by a Given Graph | Nagwa

# Question Video: Determining the Inequality Represented by a Given Graph Mathematics • First Year of Secondary School

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Which inequality has been graphed in the given figure?

04:25

### 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 thatβs 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 ππ₯ 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. Itβs 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 doesnβt matter which coordinates we designate with the π₯ one, π¦ one or π₯ two, π¦ two values. So letβs 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 isnβt 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. Letβs 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.

Itβs 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.

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