$(1,-3)$ is not a solution to the inequality $y>4x-3$. True or False?
Answer (2)
Substitute the point ( 1 , − 3 ) into the inequality 4x - 3"> y > 4 x − 3 .
Evaluate the expression: 4(1) - 3"> − 3 > 4 ( 1 ) − 3 simplifies to 1"> − 3 > 1 .
Determine if the inequality is true: 1"> − 3 > 1 is false.
Conclude that the statement " ( 1 , − 3 ) is not a solution to the inequality 4x - 3"> y > 4 x − 3 " is True .
Explanation
Understanding the Problem We are given the inequality 4x - 3"> y > 4 x − 3 and the point ( 1 , − 3 ) . We want to determine if the statement " ( 1 , − 3 ) is not a solution to the inequality 4x - 3"> y > 4 x − 3 " is true or false. To do this, we need to substitute the coordinates of the point into the inequality and check if the inequality holds.
Substituting the Point Substitute x = 1 and y = − 3 into the inequality 4x - 3"> y > 4 x − 3 . This gives us 4(1) - 3"> − 3 > 4 ( 1 ) − 3 .
Evaluating the Expression Now, we evaluate the right side of the inequality: 4 ( 1 ) − 3 = 4 − 3 = 1 . So the inequality becomes 1"> − 3 > 1 .
Checking the Inequality We need to check if 1"> − 3 > 1 is a true statement. Since − 3 is less than 1 , the statement 1"> − 3 > 1 is false.
Conclusion Since the inequality 1"> − 3 > 1 is false, the point ( 1 , − 3 ) is not a solution to the inequality 4x - 3"> y > 4 x − 3 . Therefore, the statement " ( 1 , − 3 ) is not a solution to the inequality 4x - 3"> y > 4 x − 3 " is true.
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, consider a budget constraint where you have a limited amount of money to spend on different items. The inequality helps define the feasible region of spending. Similarly, in manufacturing, inequalities are used to set quality control standards, ensuring that products meet certain specifications. In optimization problems, inequalities define the constraints within which a solution must lie, guiding decision-making to achieve the best possible outcome.
The point ( 1 , − 3 ) is not a solution to the inequality 4x - 3"> y > 4 x − 3 because substituting it into the inequality results in a false statement. The inequality evaluates to 1"> − 3 > 1 , which is false. Therefore, the statement is true.
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