Problem 63 - Algebra 1

Question

Check whether each ordered pair is a solution of the inequality. $$3 x-2 y<2 ;(1,3),(2,0)$$

Step-by-Step Solution

Answer

The ordered pair (1,3) is a solution of the given inequality but the ordered pair (2,0) is not a solution.

Step 1: Substitute the Pair (1,3)

First, we will substitute the values of (x, y) from the first ordered pair into the inequality. Thus, putting x=1 and y=3 into our inequality \(3x - 2y < 2\), we get \(3*1 - 2*3 < 2\).

Step 2: Evaluate the Inequality for (1, 3)

Now, evaluating the inequality, it becomes -3 < 2, which is true. Thus, (1,3) is a solution for the given inequality.

Step 3: Substitute the Pair (2,0)

Next, we will substitute the values of (x, y) from the second ordered pair into the inequality. Thus, putting x=2 and y=0 into our inequality \(3x - 2y < 2\), we get \(3*2 - 2*0 < 2\).

Step 4: Evaluate the Inequality for (2, 0)

Now, evaluating the inequality, it becomes 6 < 2, which is not true. Thus, (2,0) is not a solution for the given inequality.

Key Concepts

Understanding Ordered Pairs

Ordered pairs are a fundamental concept in algebra that provide a way to pair two numbers together. These pairs are often noted as

• (x, y), where 'x' is the first element, usually referring to the horizontal position on the Cartesian plane,
• 'y' is the second element, referring to the vertical position.

Each component in an ordered pair has a specific role, with 'x' being the input value and 'y' being the corresponding output. To determine if an ordered pair is a solution when substituted into an inequality, both the x and y values are individually replaced into the equation. The relationship dictated by the inequality symbol decides if the ordered pair satisfies the condition.

Finding a Solution of an Inequality

An inequality is similar to an equation, but instead of showing that two expressions are equal, it shows that one is greater or smaller than another. A solution of an inequality is essentially a set of values for the variables that make the inequality true. When checking for a solution, you substitute the respective ordered pair into the inequality. Then

• replace each variable with the numbers from the ordered pair,
• perform the algebraic operations following the standard arithmetic rules.

If the resulting statement holds true (i.e., follows the inequality sign's direction), the ordered pair is a solution. For instance, when checking the pair (1, 3) in the inequality, the evaluation leads to –3 < 2, which is true, confirming (1, 3) as a solution.

Evaluating Inequalities

Evaluating inequalities involves simulating the conditions set by the inequality with numerical values in place of variables. You start this process by replacing the variables with the numbers from the ordered pair in question to see if the inequality holds. Once the substitution is done, the aim is to simplify both sides of the inequality as much as possible. It's a bit like solving regular equations with some differences: inequalities express a range of possible answers instead of a single solution. During this process:

• Complete any multiplication or division first, following the order of operations (PEMDAS—Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).
• Perform addition or subtraction afterwards.
• Finally, compare the simplified expressions with the inequality sign.

If the inequality holds true after all operations, the ordered pair is a valid solution. For example, when substituting (2, 0), the inequality yields 6 < 2, which is false, indicating that (2, 0) is not a solution.