Question

On a coordinate plane, a dashed straight line with a negative slope goes through (0, 5) and (5, 0). Everything to the left of the line is shaded.Audrey has x pounds of red grapes and y pounds of green grapes. She has less than 5 pounds of grapes in all.Which are reasonable solutions for this situation? Check all that apply.(–1, 2)(1, 3.5)(2, 2)(4.5, 0.5)(5, 0)

Answer 1

To determine which points are reasonable solutions to the situation described, we need to analyze the constraints.

1. Understanding the Line: The line passes through the points (0, 5) and (5, 0). The equation of this line can be derived from its slope-intercept form (y = mx + b):

   • The slope (m) between the two points can be calculated as: $$m=\frac{y_2-y_1}{x_2-x_1}=\frac{0-5}{5-0}=-1$$
   • The y-intercept (b) is 5, since the line crosses the y-axis at (0, 5).

   Therefore, the equation of the line is: $$y=-x+5$$

2. Inequality: Since everything to the left of the line is shaded and represents solutions, we want: $$y<-x+5$$

3. Weight Constraint: Audrey has less than 5 pounds of grapes in total, which leads to the constraint: $$x+y<5$$

Now we will consider each point given and see if it meets both $y<-x+5$ and $x+y<5$:

1. Point (-1, 2):

   • $y<-x+5$: $2<-(-1)+5$ → $2<6$ (True)
   • $x+y<5$: $-1+2<5$ → $1<5$ (True)
   • Valid point.

2. Point (1, 3.5):

   • $y<-x+5$: $3.5<-1+5$ → $3.5<4$ (True)
   • $x+y<5$: $1+3.5<5$ → $4.5<5$ (True)
   • Valid point.

3. Point (2, 2):

   • $y<-x+5$: $2<-2+5$ → $2<3$ (True)
   • $x+y<5$: $2+2<5$ → $4<5$ (True)
   • Valid point.

4. Point (4.5, 0.5):

   • $y<-x+5$: $0.5<-4.5+5$ → $0.5<0.5$ (False, it is not less than)
   • $x+y<5$: $4.5+0.5<5$ → $5<5$ (False)
   • Not a valid point.

5. Point (5, 0):

   • $y<-x+5$: $0<-5+5$ → $0<0$ (False)
   • $x+y<5$: $5+0<5$ → $5<5$ (False)
   • Not a valid point.

Summary of Valid Points

The reasonable solutions based on the conditions are:

• (-1, 2)
• (1, 3.5)
• (2, 2)

So the valid points are: $(-1,2),(1,3.5),(2,2)$.

Answer 2

(1, 3.5)
(2, 2)

A graph shows number of binders on the x-axis, from 0 to 10, and number of notebooks on the y-axis, from 0 to 10. A solid straight line has a negative slope and goes through (0, 10) and (5, 0). Everything to the left of the line is shaded.Eula needs to buy binders that cost $4 each and notebooks that cost $2 each. She has $20. The graph of the inequality 4x + 2y ≤ 20, which represents the situation, is shown.What is the greatest number of binders Eula can buy? What is the greatest number of notebooks Eula can buy? If Eula buys 7 notebooks, what is the greatest number of binders she can buy?

Answer 3

To solve this problem, we need to analyze the constraints given by Eula's budget and the inequality representing her spending.

Step 1: Understanding the Inequality

The inequality that represents Eula's budget is: $$4x+2y\le 20$$ where:

• $x$ = number of binders
• $y$ = number of notebooks

Step 2: Rearranging the Inequality

Let's rearrange the inequality to form a more standard linear equation for graphing: $$2y\le 20-4x$$ $$y\le 10-2x$$

Step 3: Identify the Boundary Line

The boundary line, where $4x+2y=20$, intersects the axes:

• When $x=0$: $$4(0)+2y=20\Rightarrow 2y=20\Rightarrow y=10$$

• When $y=0$: $$4x+2(0)=20\Rightarrow 4x=20\Rightarrow x=5$$

Thus, the line intersects the y-axis at $(0,10)$ and the x-axis at $(5,0)$.

Step 4: Maximize Binders and Notebooks

• Greatest Number of Binders: To find the maximum number of binders ($x$), we should set $y=0$: $$4x+2(0)=20\Rightarrow 4x=20\Rightarrow x=5$$

So, the greatest number of binders Eula can buy is 5.

• Greatest Number of Notebooks: To find the maximum number of notebooks ($y$), we should set $x=0$: $$4(0)+2y=20\Rightarrow 2y=20\Rightarrow y=10$$

So, the greatest number of notebooks Eula can buy is 10.

Step 5: If Eula Buys 7 Notebooks

If Eula buys 7 notebooks ($y=7$), we substitute this value back into the inequality: $$4x+2(7)\le 20$$ $$4x+14\le 20$$ $$4x\le 20-14$$ $$4x\le 6$$ $$x\le \frac{6}{4}=1.5$$

Since Eula cannot buy a fraction of a binder, the maximum whole number of binders she can buy is 1 if she buys 7 notebooks.

Summary of Answers

1. The greatest number of binders Eula can buy: 5
2. The greatest number of notebooks Eula can buy: 10
3. If Eula buys 7 notebooks, the greatest number of binders she can buy: 1.

Answer 4

In a beanbag toss game, Janelle scores 5 points for landing on a round target and 8 points for landing on a square target. She needs more than 50 points to win. Let x represent the number of times Janelle lands on the round target and let y represent the number of times she lands on the square target. Which inequality represents the situation?

5x + 8y > 50
5x + 8y ≥ 50
8x + 5y > 50
8x + 5y < 50

Answer 5

To determine the correct inequality representing the situation where Janelle scores points based on her tosses, we need to analyze the scoring system:

• Points for round target: $5$ points for each time she lands on it, represented by $x$.
• Points for square target: $8$ points for each time she lands on it, represented by $y$.

Janelle needs more than 50 points to win. This can be expressed mathematically as an inequality.

Given the scoring system:

• The total points scored can be represented as: $$5x+8y$$

Since Janelle needs more than 50 points, the correct inequality would be: $$5x+8y>50$$

So, the inequality that represents the situation is: $$\boxed{{\displaystyle 5x+8y>50}}$$

Answer 6

Which is the graph of the solution set of 5x + 8y > 50? 1. On a coordinate plane, a dashed straight line with a negative slope goes through (0, 6) and (10, 0). Everything below and to the left of the line is shaded. 2. On a coordinate plane, a solid straight line with a negative slope goes through (0, 6) and (10, 0). Everything above and to the right of the line is shaded. 3. On a coordinate plane, a dashed straight line with a negative slope goes through (0, 6) and (10, 0). Everything above and to the right of the line is shaded.

Answer 7

To determine the correct graph for the inequality $5x+8y>50$, we first need to analyze the corresponding equation $5x+8y=50$.

Step 1: Find the intercepts of the equation

To graph the line represented by the equation $5x+8y=50$:

1. Find the y-intercept (set $x=0$): $$5(0)+8y=50⟹8y=50⟹y=6.25$$

2. Find the x-intercept (set $y=0$): $$5x+8(0)=50⟹5x=50⟹x=10$$

Thus, the line intersects the y-axis at $(0,6.25)$ and the x-axis at $(10,0)$.

Step 2: Determine the slope

The slope of this line is negative, which means it will decline as $x$ increases.

Step 3: Identify the shading for the inequality

Since the inequality is $5x+8y>50$, we will:

• Graph the line as dashed because the inequality does not include equality (it's strictly greater than).
• Shade above the line because we want the values of $y$ that make the left side (i.e., $5x+8y$) greater than $50$.

Step 4: Analyze the options provided

1. Option 1: Dashed line with shading below and to the left. (Incorrect, shading should be above.)
2. Option 2: Solid line with shading above and to the right. (Incorrect, it should be a dashed line, not solid.)
3. Option 3: Dashed line with shading above and to the right. (Correct.)

Conclusion

The correct graph of the solution set for the inequality $5x+8y>50$ is:

Option 3: A dashed line with shading above and to the right of the line.

Answer 8

A graph shows Round Target on the x-axis, from 0 to 10, and Square Target on the y-axis, from 0 to 10. A dashed straight line has a negative slope and goes through (2, 5) and (10, 0). Everything above and to the right of the line is shaded.The graph represents the ways Janelle can win the beanbag toss game. Which describes a way Janelle can win the game?land on round 7 times; land on square 1 timeland on round 6 times; land on square 2 timesland on round 5 times; land on square 3 timesland on round 4 times; land on square 4 times

Answer 9

To determine which option describes a way Janelle can win the beanbag toss game based on the given graph, we need to evaluate the scoring for each option in relation to the inequality represented by the graph.

Understanding the Score

Janelle scores:

• 5 points for each round target (denoted as $x$)
• 8 points for each square target (denoted as $y$)

Given the inequality $5x+8y>50$, we will check each option to see if it satisfies this inequality.

Scoring Calculation for Each Option:

1. Option 1: Land on round 7 times; land on square 1 time. $$5(7)+8(1)=35+8=43(43\le 50)\text{(Not a winning score)}$$

2. Option 2: Land on round 6 times; land on square 2 times. $$5(6)+8(2)=30+16=46(46\le 50)\text{(Not a winning score)}$$

3. Option 3: Land on round 5 times; land on square 3 times. $$5(5)+8(3)=25+24=49(49\le 50)\text{(Not a winning score)}$$

4. Option 4: Land on round 4 times; land on square 4 times. $$5(4)+8(4)=20+32=52(52>50)\text{(Winning score)}$$

Conclusion

The only option that satisfies $5x+8y>50$ is:

Option 4: Land on round 4 times; land on square 4 times.

This would be a way Janelle can win the beanbag toss game.

Answer 10

Which shows the graph of the solution set of 6x + 4y < 12? 1. On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 3) and (3, 1). Everything below and to the left of the line is shaded. 2. On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 3) and (3, 1). Everything above and to the right of the line is shaded. 3. On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 3) and (2, 0). Everything to the left of the line is shaded. 4. On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 3) and (2, 0). Everything to the right of the line is shaded.

Answer 11

To determine which graph accurately represents the solution set of the inequality $6x+4y<12$, we first need to understand the corresponding equation and the behavior of the inequality.

Step 1: Find the Boundary Line

The inequality can be rearranged to form the equation: $$6x+4y=12$$

To graph this line, we can find the intercepts.

1. Find the y-intercept (set $x=0$): $$6(0)+4y=12⟹4y=12⟹y=3$$ So, the y-intercept is $(0,3)$.

2. Find the x-intercept (set $y=0$): $$6x+4(0)=12⟹6x=12⟹x=2$$ So, the x-intercept is $(2,0)$.

Step 2: Determine the Inequality

Since we are given the inequality $6x+4y<12$, we will:

• Graph the line as dashed because the inequality is strict (not ≤).
• Shade below the line because we want the values of $y$ that make $6x+4y$ less than $12$.

Step 3: Review Graph Options

Now let's analyze each option given:

1. Option 1: Dashed line through $(0,3)$ and $(3,1)$ with shading below and to the left. (Incorrect) - The x-intercept is not correct as it states $(3,1)$.

2. Option 2: Dashed line through $(0,3)$ and $(3,1)$ with shading above and to the right. (Incorrect) - Should be shading below.

3. Option 3: Dashed line through $(0,3)$ and $(2,0)$, with shading to the left. (Correct) - This correctly represents the line and the correct shading.

4. Option 4: Dashed line through $(0,3)$ and $(2,0)$ with shading to the right. (Incorrect) - Should be shading below the line.

Conclusion

The correct graph of the solution set of $6x+4y<12$ is:

Option 3: A dashed line through $(0,3)$ and $(2,0)$ with shading below and to the left of the line.