The point $(0,0)$ is a solution to which of these inequalities?
A. $y-6<2 x-7$
B. $y+7<2 x+6$
C. $y+7<2 x-6$
D. $y-7<2 x-6$
Answer (1)
Substitute ( 0 , 0 ) into inequality A: − 6 < − 7 , which is false.
Substitute ( 0 , 0 ) into inequality B: 7 < 6 , which is false.
Substitute ( 0 , 0 ) into inequality C: 7 < − 6 , which is false.
Substitute ( 0 , 0 ) into inequality D: − 7 < − 6 , which is true. Therefore, the answer is D .
Explanation
Problem Analysis We are given the point ( 0 , 0 ) and asked to determine which of the given inequalities has this point as a solution. To do this, we will substitute x = 0 and y = 0 into each inequality and check if the resulting statement is true.
Testing Inequality A A. y − 6 < 2 x − 7 Substituting x = 0 and y = 0 , we get: 0 − 6 < 2 ( 0 ) − 7 − 6 < − 7 This statement is false, since − 6 is greater than − 7 .
Testing Inequality B B. y + 7 < 2 x + 6 Substituting x = 0 and y = 0 , we get: 0 + 7 < 2 ( 0 ) + 6 7 < 6 This statement is false, since 7 is greater than 6 .
Testing Inequality C C. y + 7 < 2 x − 6 Substituting x = 0 and y = 0 , we get: 0 + 7 < 2 ( 0 ) − 6 7 < − 6 This statement is false, since 7 is greater than − 6 .
Testing Inequality D D. y − 7 < 2 x − 6 Substituting x = 0 and y = 0 , we get: 0 − 7 < 2 ( 0 ) − 6 − 7 < − 6 This statement is true, since − 7 is less than − 6 .
Conclusion Therefore, the point ( 0 , 0 ) is a solution to the inequality y − 7 < 2 x − 6 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, when planning a budget, you might use inequalities to determine how much you can spend on different categories while staying within your income. If your income is I and you want to allocate amounts x , y , and z to housing, food, and entertainment respectively, the inequality x + y + z ≤ I ensures you don't overspend. Similarly, in manufacturing, inequalities can help ensure that products meet certain quality standards, such as weight or dimensions, by setting acceptable ranges.
