The solutions to the inequality $y>-3 x+2$ are shaded on the graph. Which point is a solution?
(0,2)
(2,0)
(1,-2)
(-2,1)
Answer (2)
To find a solution to the inequality -3x + 2"> y > − 3 x + 2 , we tested four points. Only the point ( 2 , 0 ) satisfies the inequality, making it the solution. Therefore, the answer is ( 2 , 0 ) .
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Substitute each point into the inequality -3x + 2"> y > − 3 x + 2 .
Check if the inequality holds true for each point.
Point ( 0 , 2 ) : -3(0) + 2 \Rightarrow 2 > 2"> 2 > − 3 ( 0 ) + 2 ⇒ 2 > 2 (False).
Point ( 2 , 0 ) : -3(2) + 2 \Rightarrow 0 > -4"> 0 > − 3 ( 2 ) + 2 ⇒ 0 > − 4 (True).
Point ( 1 , − 2 ) : -3(1) + 2 \Rightarrow -2 > -1"> − 2 > − 3 ( 1 ) + 2 ⇒ − 2 > − 1 (False).
Point ( − 2 , 1 ) : -3(-2) + 2 \Rightarrow 1 > 8"> 1 > − 3 ( − 2 ) + 2 ⇒ 1 > 8 (False).
The point ( 2 , 0 ) is a solution: ( 2 , 0 ) .
Explanation
Understanding the Problem We are given the inequality -3x + 2"> y > − 3 x + 2 and four points: ( 0 , 2 ) , ( 2 , 0 ) , ( 1 , − 2 ) , and ( − 2 , 1 ) . We need to determine which of these points satisfies the inequality.
Testing the Points Let's test each point to see if it satisfies the inequality.
Testing (0,2)
Point ( 0 , 2 ) : Substitute x = 0 and y = 2 into the inequality: -3(0) + 2"> 2 > − 3 ( 0 ) + 2 0 + 2"> 2 > 0 + 2 2"> 2 > 2 This is false, so ( 0 , 2 ) is not a solution.
Testing (2,0)
Point ( 2 , 0 ) : Substitute x = 2 and y = 0 into the inequality: -3(2) + 2"> 0 > − 3 ( 2 ) + 2 -6 + 2"> 0 > − 6 + 2 -4"> 0 > − 4 This is true, so ( 2 , 0 ) is a solution.
Testing (1,-2)
Point ( 1 , − 2 ) : Substitute x = 1 and y = − 2 into the inequality: -3(1) + 2"> − 2 > − 3 ( 1 ) + 2 -3 + 2"> − 2 > − 3 + 2 -1"> − 2 > − 1 This is false, so ( 1 , − 2 ) is not a solution.
Testing (-2,1)
Point ( − 2 , 1 ) : Substitute x = − 2 and y = 1 into the inequality: -3(-2) + 2"> 1 > − 3 ( − 2 ) + 2 6 + 2"> 1 > 6 + 2 8"> 1 > 8 This is false, so ( − 2 , 1 ) is not a solution.
Conclusion Only the point ( 2 , 0 ) satisfies the inequality -3x + 2"> y > − 3 x + 2 . Therefore, ( 2 , 0 ) is a solution.
Examples
Understanding inequalities is crucial in many real-world scenarios. For example, when planning a budget, you might use an inequality to ensure that your expenses are less than or equal to your income. Similarly, in business, companies use inequalities to model constraints on resources and production to maximize profit. Inequalities also play a vital role in optimization problems, such as determining the most efficient way to allocate resources or minimize costs.
