Which of the regions on the graph contain solutions to the inequality 𝑦 is greater than or equal to two 𝑥 minus four?
When reading the question, it is important to note that the word “regions” is plural, which suggests there is more than one region that satisfies the inequality. We can begin this question by considering which line on our graph corresponds to the equation 𝑦 is equal to two 𝑥 minus four. Any linear equation which corresponds to a straight line graph can be written in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope or gradient and 𝑏 is the 𝑦-intercept. Our equation has a slope or gradient of two and a 𝑦-intercept of negative four. This means that our graph must cross the 𝑦-axis at negative four.
A positive slope or gradient means that our line must slope up from left to right. As the slope is equal to two, for every one unit we move to the right, we must move two units up. The line that corresponds to the equation 𝑦 equals two 𝑥 minus four is shown in pink. As the inequality sign is greater than or equal to, we draw a solid line, whereas had it been strictly greater than or strictly less than, we would’ve drawn a dashed or broken line. As the 𝑦-values need to be greater than or equal to two 𝑥 minus four, the region must lie above our line. There are four such regions on our figure, regions A, B, C, and D. The regions on the graph that contain solutions to the inequality 𝑦 is greater than or equal to two 𝑥 minus four are A, B, C, and D.
We can check this answer by substituting in the coordinates of any points in these regions. For example, the point with coordinates negative five, five lies in region A. The 𝑥-value here is negative five, and the 𝑦-value is five. Substituting these values into our inequality, we have five is greater than or equal to two multiplied by negative five minus four. Two multiplied by negative five is negative 10, and subtracting four from this gives us negative 14. As five is greater than or equal to negative 14, this point does satisfy our inequality. We could repeat this process for any other point in the regions A, B, C, and D.