Problem 53 - Algebra 1: Concepts and Skills

Question

Decide whether the ordered pair is a solution of the inequality. $$ y \geq x^{2}-25 ;(5,5) $$

Step-by-Step Solution

Answer

Yes, the ordered pair (5,5) is a solution to the inequality.

Step 1: Substitute Values

The first step in solving this problem involves substituting the values of the ordered pair for the variables x and y in the equation. In this case, x = 5 and y = 5. Substitute these values into the inequality: \(5 \geq 5^{2}-25\)

Step 2: Evaluate

The next step is to simplify the inequality to check if it is true. Start by performing the operation in the parenthesis: \(5^{2}\) is \(=25\), giving \(5 \geq 25-25\).

Step 3: Verify

After the terminology, the equation simplifies to \(5 \geq 0\). Since 5 is indeed greater than 0, this validates the ordered pair of (5,5). The inequality holds true, and thus the ordered pair is a solution.

Key Concepts

Understanding Ordered Pairs

In mathematics, an ordered pair is a set of two elements where the order in which these elements appear is significant. Ordered pairs are typically used to denote coordinates on a graph, such as \((x, y)\). The first element of the pair, \(x\), is often called the "abscissa," and the second element, \(y\), is called the "ordinate."

• Abscissa (x) - The horizontal value.
• Ordinate (y) - The vertical value.

Understanding ordered pairs is essential because they help in plotting points on a graph and allow us to solve systems of equations or inequalities. When given an ordered pair, it’s important to ensure that the values are substituted correctly into any given equation or inequality to determine their truth.

Substitution in Inequalities

Substitution is a method used to replace variables with specific values in mathematical expressions. This technique is crucial when working with inequalities to test potential solutions. In our example, given the ordered pair \((5, 5)\), substitution involves inserting these values into the inequality \(y \geq x^2 - 25\).

By replacing \(x\) with 5 and \(y\) with 5, the inequality becomes \(5 \geq 5^2 - 25\). This step transforms an abstract inequality into a concrete statement that can be evaluated. Accurate substitution is key to correctly solving or proving inequalities.

Evaluating Inequality

Once substitution has been performed, the next step is to evaluate the inequality. This means simplifying the mathematical expression to verify if it holds true. In the example provided, we've already substituted and have:\[5 \geq 5^2 - 25\].

Now, calculate the square of 5, which is 25, thus the inequality simplifies to:

• Calculate \(5^2\), getting 25.
• Subtract 25 from 25, resulting in 0.
• Replace into inequality: \(5 \geq 0\).

The resulting statement decides whether the given ordered pair (5,5) satisfies the inequality. The simplification process is fundamental to clearly understand if and how an inequality holds true.

Verifying a Solution

Verification involves confirming that the simplified statement derived from substitution is valid. Essentially, you verify a solution by ensuring that the mathematical inequality is satisfied when evaluated with the given numbers. In this instance, we ended up with the inequality:

\(5 \geq 0\).

• Since 5 is greater than 0, the inequality holds.
• The ordered pair \((5, 5)\) is indeed a solution.

Verification solidifies our confidence in the chosen solution. The process acts as a double-check, proving that substitution and evaluation were performed correctly, assuring that the answer is logically sound and correctly reasoned.