Problem 88 - Algebra and Trigonometry

Question

How do you determine if an ordered pair is a solution of an inequality in two variables, \(x\) and \(y ?\)

Step-by-Step Solution

Answer

To determine if an ordered pair is a solution of an inequality, the x and y values from the ordered pair are substituted into the inequality and it is checked to see if the inequality holds true. If it does, then the pair is a solution. If it does not, then the pair is not a solution.

Step 1: Understand the elements in the exercise

Remember that an ordered pair is a pair of mathematical objects for which the order of the elements is significant. They are generally written in the form (x, y), where 'x' is the first element and 'y' is the second element. The ordered pair is a solution to the inequality if, when the x and y values of the ordered pair are substituted into the inequality, the inequality holds true.

Step 2: Substitute the ordered pair into the inequality

Given an inequality, say for instance \(3x - 2y > 0\), and an ordered pair, say (1,2), plug in the x and y values from the ordered pair into the inequality. This means replacing 'x' with '1' and 'y' with '2'. So the inequality becomes \(3(1) - 2(2) > 0\)

Step 3: Investigate the truthiness of the inequality

Evaluate the inequality to ascertain whether it holds true or not. \(3(1) - 2(2)\) simplifies to \(3 - 4\), which is \(-1\). Since \(-1\) is not greater than \(0\), the inequality does not hold true, thus the ordered pair (1,2) is not a solution to the inequality \(3x - 2y > 0\)