Problem 18 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the inequality. $$ y<5 x^{2}+8,(3,45) $$

Step-by-Step Solution

Answer

Yes, the ordered pair (3,45) is a solution to the inequality \( y < 5x^{2} + 8 \).

Step 1 - Substitution

Firstly, substitute the values from the ordered pair into the inequality. So, replace \(x\) with 3 and \(y\) with 45. Therefore, the inequality becomes \(45 < 5(3)^{2} + 8 \).

Step 2 - Simplifying the inequality

Simplify the right-hand side by first dealing with the exponent: \(3^{2}=9\). Then, multiply 5 by 9 to get 45, and finally add 8. The inequality simplifies to \(45 < 45 + 8 \) or \(45 < 53\) .

Step 3 - Verifying the inequality

Finally, verify the inequality. In this case, 45 is indeed less than 53, so the inequality holds true.

Key Concepts

Inequality Solutions

When faced with an inequality like \( y < 5x^2 + 8 \), the goal is to check if a given ordered pair, say \((x, y) = (3, 45)\), is a solution. This process is critical because it tells us if these values of \(x\) and \(y\) satisfy the inequality condition. Ordered pairs consist of two numbers, usually represented as \((x, y)\) on a Cartesian plane.To determine if the pair is a solution:

• Substitute \(x\) and \(y\) into the inequality.
• Evaluate the inequality to see if the statement is true.

For our example, we set \(y = 45\) and \(x = 3\) into the inequality and simplify to see if the left side (\(45\)) is indeed less than the right side. If it is, then the ordered pair is a solution.

Substitution Method

The substitution method simplifies the process of testing an ordered pair in an inequality or equation. It involves replacing the variables in the inequality with the corresponding values from the ordered pair.Here's how it works:

• Identify the values for \(x\) and \(y\) from the ordered pair. For instance, \(x = 3\) and \(y = 45\).
• Substitute these values directly into the inequality. For our example, the inequality \(y < 5x^2 + 8\) becomes \(45 < 5(3)^2 + 8\).

Substitution is straightforward but very powerful because it transforms a theoretical inequality into a manageable arithmetic problem, which can then be simplified to verify if the inequality holds true with the given numbers. It turns abstract math into a concrete task that can be solved step-by-step.

Simplifying Inequalities

Simplifying an inequality is all about performing arithmetic operations to evaluate the correctness of the inequality from the substitution step. This involves:

• Handling exponents first: \(3^2 = 9\).
• Multiplying the result: \(5 \times 9 = 45\).
• Add any constants: \(45 + 8 = 53\).

With these steps completed, you'll find the simplified form of the right-hand side of the inequality. For instance, after substituting \(x = 3\) into the inequality, the expression simplifies to \(45 < 53\).This process allows you to clearly determine if \(45\) is indeed less than \(53\), thereby concluding that the original inequality is valid for the given ordered pair \((3, 45)\). Simplifying inequalities helps to see the relationship between numbers much clearer and makes the problem easier to visualize. This ensures an accurate and efficient way of solving and verifying inequalities.