Which inequality has been graphed in the given figure?
The graph drawn contains a straight line, which means that our equation will be linear and in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑏 is the 𝑦-intercept and 𝑚 is the slope or gradient. As the line was bold, we know that the shaded region will represent the area that is greater than or equal to or less than or equal to. A broken or dotted line represents an inequality that is strictly greater than or strictly less than. We can see from the graph that our line crosses the 𝑦-axis at zero, three. Therefore, our value of 𝑏 is three.
By selecting another point on the line — for example, four, two — we can calculate the slope or gradient. The slope is equal to the change in the 𝑦-coordinates divided by the change in the 𝑥-coordinates. This is sometimes referred to as the rise over the run. Substituting in zero, three for 𝑥 one, 𝑦 one and four, two for 𝑥 two, 𝑦 two, we have two minus three divided by four minus zero. This is equal to negative one over four or negative one-quarter. The straight line drawn in the figure represents the equation 𝑦 is equal to negative one-quarter 𝑥 plus three.
As the shaded region is above this line, we know that the correct inequality is 𝑦 is greater than or equal to negative one-quarter 𝑥 plus three. We could check this answer by substituting in the coordinates of any point in the shaded region, for example, the point four, three. When we substitute 𝑥 equals four and 𝑦 equals three into our inequality, we get three is greater than or equal to negative a quarter multiplied by four plus three. The right-hand side simplifies to two. And as three is greater than two, it satisfies the inequality 𝑦 is greater than or equal to negative a quarter 𝑥 plus three.