The graph shows all pairs that satisfy a constraint for a club’s weekly participation goals, where is the number of hours spent on planning and is the number of hours spent on outreach. Which choice gives an inequality that represents the shaded region on the graph?

Solution available with step-by-step explanation

SAT Linear Inequalities Question 71 | Free SAT Prep | Aniko

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Question 71·Medium·Linear Inequalities in One or Two Variables

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The graph shows all pairs $(x,y)$ that satisfy a constraint for a club’s weekly participation goals, where $x$ is the number of hours spent on planning and $y$ is the number of hours spent on outreach.

Which choice gives an inequality that represents the shaded region on the graph?

For a graphed linear inequality, first determine the boundary line equation using two points on the line (slope and intercept). Then use the shading to decide whether the inequality is “greater than” or “less than,” and use the line style: dashed means strict ( $>$ or $<$ ) and solid means inclusive ( $\ge$ or $\le$ ). Finally, rewrite into the form that matches one of the answer choices.

Hints

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Use the two labeled points

Find the slope of the boundary line using $(0,4)$ and $(6,1)$ .

Write the line equation

Use the point with $x=0$ to identify the $y$ -intercept, then write the line in slope-intercept form.

Decide between $>$ , $<$ , $\ge$ , and $\le$

Shading above the line means $y$ is greater than the line. A dashed boundary means the line itself is not included (strict inequality).

Desmos Guide

Graph the boundary line

In Desmos, enter y=-1/2 x + 4 and confirm it passes through $(0,4)$ and $(6,1)$ .

Test the shaded side

Try a point clearly in the shaded region (for example, $(0,6)$ ) and check whether it makes x+2y>8 true.

Match the inequality to the shading

Enter x+2y>8 in Desmos and confirm the shaded region matches the graph; the correct inequality is $x+2y>8$ .

Step-by-step Explanation

Find the boundary line equation

The dashed line passes through $(0,4)$ and $(6,1)$ . Its slope is

\frac{1-4}{6-0}=-\frac{3}{6}=-\frac{1}{2}.

Since the line goes through $(0,4)$ , the $y$ -intercept is $4$ , so the boundary line is

y=-\frac{1}{2}x+4.

Use shading and line type to choose the inequality

The region above the line is shaded, so $y$ must be greater than the line’s value.

Because the boundary is dashed, points on the line are not included, so the inequality is strict:

y>-\frac{1}{2}x+4.

Rewrite in standard form and select the choice

Multiply by $2$ to clear the fraction:

2y>-x+8.

Add $x$ to both sides:

x+2y>8.

Therefore, the correct choice is $x+2y>8$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation