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Chapter 9: Problem 66

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

Short Answer

Expert verified
An ordered pair is a solution to a given inequality in two variables, x and y, if when we substitute the x and y values from the pair into the inequality, the inequality holds true. For our evaluation, the pair (2, 3) is not a solution to the inequality \(3x + 2y < 12\), since swapping x and y into the inequality results in an untrue statement, \(12 < 12\).

Step by step solution

01

Interpret the ordered pair and inequality

Firstly, remember that an ordered pair takes the form (x, y). Inequalities are mathematical statements indicating how two values compare to each other, using symbols like <, >, \(\leq\), or \(\geq\). For example, if we have an inequality \(3x + 2y < 12\), we're saying that whatever values x and y take, when multiplied by their respective coefficients (3 for x, 2 for y), and then added together, the result must be less than 12.
02

Substitute the ordered pair into the inequality

The next step is to take the ordered pair and substitute the x and y values into the inequality. If the ordered pair we're testing is (2, 3), using the aforementioned inequality as an example, we would substitute x by 2 and y by 3. We'd get \(3(2) + 2(3) < 12\).
03

Evaluate & interpret the result

Now calculate the left side of the inequality. For our example, we calculate \(3(2) + 2(3)\) which equals 12. Our inequality now reads \(12 < 12\). Since 12 is not less than 12, it means our pair (2, 3) is not a solution to the inequality \(3x + 2y < 12\). If the inequalities hold true after plugging in the pair, then the ordered pair is a solution to the provided inequality.

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