Question 28·Easy·Linear Inequalities in One or Two Variables
The graph above shows a shaded region in the -plane that represents all solutions to a linear inequality.
Which ordered pair is a solution to this inequality?
For graph-of-inequality questions, think visually: the shaded region is the solution set. Instead of trying to reverse-engineer the exact inequality, quickly plot each answer choice by moving along the -axis, then up or down to the -value, and check whether the point lies in the shaded area or the unshaded area (and whether the boundary line is solid or dashed). Choose the option that clearly falls inside the shading; if a point is on a dashed boundary or in the unshaded region, eliminate it immediately.
Hints
Connect shading to solutions
Remember that for a graphed inequality, every point in the shaded area represents a solution, and any point in the unshaded area is not a solution.
Plot the answer choices
Try sketching where each answer choice would be on the coordinate plane: start with the -coordinate (left/right), then the -coordinate (up/down). Don’t worry about being perfect—just be accurate enough to tell which side of the boundary line the point is on.
Compare each point to the shaded region
After you place each point, ask: does this point fall in the shaded region, on the boundary line, or in the unshaded region? Only a point inside the shaded region (or on a solid boundary line, if shown) can be a solution.
Desmos Guide
Recreate the boundary line in Desmos
Look at the graph in the problem and identify two clear points that lie on the boundary line (for example, where it crosses grid intersections). Use those two points to find the line’s equation (determine its slope and intercept), then type that equation into Desmos as a line, such as y = mx + b with the values you found.
Turn the line into an inequality that matches the shading
In Desmos, change the line into an inequality (for example, y > mx + b or y ≤ mx + b) and adjust the inequality sign so that the shaded side in Desmos matches the shaded region shown in the question’s graph. This recreates the solution region of the inequality.
Plot each answer choice as a point
Type each option as a point in Desmos: (0,2), (2,3), (-4,2), and (2,0). Look at the graph and see which of these points lies inside the shaded region produced by your inequality in Desmos. The point that appears in the shaded region is the solution to the inequality.
Step-by-step Explanation
Understand what the shaded region represents
For a graph of a linear inequality, the shaded region shows all the points that make the inequality true. Any point in the unshaded region (or on a dashed boundary line that is not included) does not satisfy the inequality.
Recall how to locate a point from its coordinates
An ordered pair tells you:
- Move units left/right along the -axis (right if , left if ).
- Then move units up/down (up if , down if ) to place the point.
You will use this to mentally (or lightly with your pencil) place each answer choice on the given graph.
Test each answer choice against the shaded region
For each option:
- Plot , , , and on the coordinate grid shown in the problem.
- After you place each point, check whether it lands inside the shaded region or in the unshaded part of the graph.
You should find that three of these points land outside the shaded region (or on a part of the boundary that is not included), and exactly one lands clearly inside the shaded region.
Identify the point that is in the shaded region
When you compare the plotted points with the shading, the only point that clearly lies inside the shaded region is . Therefore, is a solution to the inequality, and choice B is correct.