Which inequality has been graphed in the given figure?
We will begin by working out the equation of the straight line shown. We know that this will correspond to an equation in slope–intercept form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑏 is the 𝑦-intercept and 𝑚 is the slope or gradient. Our line crosses the 𝑦-axis at negative five. Therefore, 𝑏 is equal to negative five. The gradient or slope can be calculated by working out the change in the 𝑦-coordinates divided by the change in the 𝑥-coordinates. This is also sometimes known as the rise over the run.
Selecting a second point on our line, in this case one, zero, we can create a right triangle which has a rise of five and a run of one. The change in the 𝑦-coordinates is equal to zero minus negative five, and the change in the 𝑥-coordinates is one minus zero. This is equal to five over one, which in turn equals five. The slope or gradient of our line is five. This means that our line has equation 𝑦 is equal to five 𝑥 minus five.
We have been asked to write the inequality in the given figure. As the initial line was bold, we know that our inequality will be greater than or equal to or less than or equal to. A dotted or broken line would mean that the inequality was strictly greater than or strictly less than. As the area shaded is below our line, the correct inequality is 𝑦 is less than or equal to five 𝑥 minus five.
We could check this answer by selecting a point in the shaded region, for example, three, two. Substituting in the values gives us two is less than or equal to five multiplied by three minus five. Five multiplied by three minus five is equal to 10. And as two is less than or equal to 10, we know that our inequality is correct. The graph in the given figure represents the inequality 𝑦 is less than or equal to five 𝑥 minus five.