A system of inequalities asks for points that satisfy every condition at once. When the choices are tables, graph the inequalities in Desmos and look for the overlapping shading. One point outside the overlap rules out its whole table, but the table you choose needs every row checked. Watch the boundaries: a strict inequality excludes points on its line.
Hints
- Hint 1
A system of inequalities requires both statements to be true for the same pair. Desmos shades the points that satisfy each statement. Where do the two shadings overlap?
- Hint 2
A table claims that every row works. You can reject it as soon as you find one pair outside the overlap. Which pair would you check in each table?
- Hint 3
A boundary line contains points that make the two sides equal. The line for is included; the line for is not. What does that mean for a point touching either line?
Step-by-step
Graph the overlap and test the tables
Step 1Find the region that satisfies both inequalities
Type both inequalities into Desmos, one per line. Desmos shades the solutions to each one. The overlap contains the pairs that satisfy both. It draws a solid boundary for , where equality counts, and a dashed boundary for , where equality doesn't count.
- Step 2
Use one failing pair to reject a table
Plot , , and on separate lines. Desmos shows that misses the first shading. The point sits on the dashed line , so it fails the strict condition. The point is outside the overlap too. One failing pair rules out its entire table.
- Step 3
Check every pair in the remaining table
Plot , , and . Desmos shows all three in the overlap. The point touches the solid boundary, so it counts; neither of the other two touches an excluded boundary. Every pair in this table satisfies both inequalities. Choice D.