Video Transcript
Which inequality has been graphed in the given figure?
We can see from the figure that our graph includes a straight line, which will be a linear equation of the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑏 corresponds to the 𝑦-intercept and 𝑚 is the slope or gradient. The graph crosses the 𝑦-a𝑥is at the point with coordinates zero, negative three. This means that the 𝑦-intercept is equal to negative three.
If we select another point on our line, for example, negative one, negative 2.8, we can calculate the slope or gradient by dividing the change in the 𝑦-coordinates by the change in the 𝑥-coordinates. 𝑚 is equal to 𝑦 two minus 𝑦 one divided by 𝑥 two minus 𝑥 one. This is often referred to as the rise over the run.
Substituting in our coordinates, we have negative 2.8 minus negative three divided by negative one minus zero. The numerator simplifies to 0.2 and the denominator to negative one. 0.2 divided by negative one is equal to negative 0.2. This is equivalent to the fraction negative one-fifth. The equation of the straight line on our figure is 𝑦 is equal to negative one-fifth 𝑥 minus three.
As the line was solid, a line will be greater than or equal to or less than or equal to. A broken or dotted line would correspond to strictly greater than or strictly less than. As the region shaded is above this line, any point in this region will have a 𝑦-value greater than or equal to negative one-fifth 𝑥 minus three. Therefore, the correct inequality is 𝑦 is greater than or equal to negative one-fifth 𝑥 minus three.
We could check this by substituting in the coordinates of a point in our shaded region, for example, one, two. Substituting in these values gives us two is greater than or equal to negative one-fifth multiplied by one minus three. The right-hand side simplifies to negative three and one-fifth, and two is greater than this. This confirms that we’ve selected the correct side of the line 𝑦 is equal to negative one-fifth 𝑥 minus three. Therefore, our inequality sign is correct.
