Question 365·Easy·Algebra

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Which of the following ordered pairs $(x, y)$ satisfies both inequalities below?

2x + y \le 5

x - y > -1

I. $(0, 5)$
II. $(2, 0)$
III. $(1, 3)$
IV. $(3, 2)$

A system of inequalities asks for a point that makes every inequality true. Type each inequality into Desmos, then plot the candidate points and look for the overlap of the shaded regions. A solid boundary counts because it comes from $\le$ or $\ge$ ; a dashed boundary does not. Don't keep a point that passes only one condition.

Hints

Hint 1
An ordered pair $(x,y)$ gives you $x$ first and $y$ second. A point must satisfy both inequalities, not merely one. Where do the two shaded regions overlap?
Hint 2
A boundary is where the two sides of an inequality are equal. The $\le$ boundary includes its points, but the $>$ boundary excludes them. Check which boundary a plotted point touches.

Step-by-step

Graph the overlap and test the points
Step 1
Graph both inequalities
Type both inequalities as written. Desmos shades the points that satisfy each one. A point satisfies both only where the shadings overlap. The $\le$ boundary is solid and included; the $>$ boundary is dashed and excluded.
Step 2
Rule out points outside the overlap
Plot $(0,5)$ , $(1,3)$ , and $(3,2)$ . Desmos shows the first two on the solid boundary but outside the other shading. The third misses the first shading. A point that passes only one inequality doesn't work.
Step 3
Check the remaining point
Plot $(2,0)$ . Desmos shows it inside both shaded regions. The numbers confirm it: $2(2)+0=4\le5$ and $2-0=2>-1$ . So $(2,0)$ satisfies both inequalities. Choice D.

Question 365·Easy·Algebra

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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