Question 470·Easy·Algebra

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A robotics competition awards $x$ points for completed challenges and subtracts $y$ penalty points for rule violations. A team’s final score is $4x-y$ .

To qualify for the next round, a team must have a final score of at least $6$ .

Which ordered pair $(x,y)$ satisfies this requirement?

A two-variable inequality describes every pair that meets a limit. The cue is a score built from $x$ and $y$ that must be at least a certain amount: the score may equal that amount or be higher. Write the inequality, graph it in Desmos, and plot the candidate pairs. Remember that penalty points are subtracted, not earned.

Hints

Hint 1
The words at least include the cutoff itself: a score of $6$ qualifies, and any higher score qualifies too. What inequality expresses that requirement for the final score $4x-y$ ?
Hint 2
An ordered pair $(x,y)$ puts its first number in $x$ and its second in $y$ . On a shaded graph, points in the shading qualify. Where do the candidate points land?

Step-by-step

Graph the qualifying scores
Step 1
Write the score requirement
The final score is $4x-y$ , and at least $6$ means $6$ or more. Write the inequality, a comparison that allows a range of scores:
$4x-y\ge6.$
Subtract the penalty points before comparing the score with the minimum.
Step 2
Graph the scores that qualify
Type $4x-y\ge6$ into Desmos. It shades the qualifying side on or below a solid line through $(2,2)$ and $(3,6)$ , matching the figure. The line is solid because a score of exactly $6$ qualifies.
Step 3
Plot the candidate pairs
Add each choice as a point in Desmos: $(1,2)$ , $(2,4)$ , $(3,1)$ , and $(0,5)$ . Only $(3,1)$ lies in the shading, so that pair gives a qualifying final score. Choice C.

Question 470·Easy·Algebra

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