Question

Decide whether the ordered pair is a solution of the inequality. $$y\geq 4 x^{2}-64 x+92 ;(1,30)$$

Short answer

No, the ordered pair (1,30) is not a solution to the inequality \(y \geq 4x^{2} - 64x + 92\).

Step-by-step solution

Understanding Inequalities

Inequalities are relations between expressions that may not be equal. It's a comparison of two values. When you substitute a pair (x,y) into inequality and after computation, if you get a true statement, that means the pair is a solution.

Substitute the Values of the Ordered Pair into the Inequality

Our given ordered pair is (1, 30) which means x=1 and y=30. Substitute these values into the inequality and evaluate: \(30 \geq 4*(1)^{2}-64*1+92\). This simplifies to \(30 \geq 32\).

Evaluate the Inequality

Having performed the substitution and simplification, we can see that the statement, \(30 \geq 32\), is false because 30 is not greater than or equal to 32.

Key concepts

Ordered Pairs as Solutions

When solving inequalities, we sometimes need to determine if an ordered pair, which consists of values for variables typically labeled as (x, y), is a solution to the inequality. An ordered pair is a solution to an inequality if, when the x and y values are plugged into the inequality, the inequality holds true.

For example, let's consider the inequality from the exercise, \(y \geq 4 x^{2}-64 x+92\). If we are given the ordered pair (1, 30), we can test if it is a solution by substituting in these values for x and y, respectively. This process will either validate or debunk the pair as a true solution of the given inequality.

An easy way to visualize this concept is to remember that each ordered pair represents a point on the coordinate plane. If this point lies on the boundary of or above the region defined by the inequality (for \(y \geq \) inequalities), then the ordered pair is a solution.

Substitution Method

The substitution method is crucial when solving for unknown variables, especially in systems of equations or inequalities. This method involves replacing variables with their corresponding values to simplify expressions or solve problems.

Following the step by step solution, in the substitution step, we take our ordered pair (1, 30) and replace 'x' with 1 and 'y' with 30 in the original inequality. It looks like this: \(30 \geq 4*(1)^{2}-64*1+92\). By performing these replacements, the inequality becomes an algebraic expression that we can solve. It is a direct and often straightforward process that hinges on basic algebraic principles.

Why the Substitution Method is Effective

The substitution method allows us to pinpoint specific solutions to the inequality by reducing abstract variables to tangible numbers. This clarifies whether the initially complex-looking inequality is actually true or not, given the specific values from the ordered pair.

Inequality Evaluation

After using the substitution method, we arrive at the stage of inequality evaluation. This step determines whether the resulting statement after the substitution is true or false. For the given inequality, after substituting the x and y values from our ordered pair, our inequality simplifies to \(30 \geq 4*(1)^{2}-64*1+92\), which further simplifies to \(30 \geq 32\).

The purpose of inequality evaluation is to compare the values on each side of the inequality symbol. If the statement aligns with the inequality's logic – in this case, checking if the left side is indeed greater than or equal to the right side – then the statement is true, and the ordered pair is a solution. However, since 30 is not greater than 32, the statement is false. Therefore, the ordered pair (1, 30) does not satisfy the inequality \(y \geq 4 x^{2}-64 x+92\).

Significance of Inequality Evaluation

Evaluating inequalities is a fundamental skill in mathematics that helps us understand relationships between quantities and solve for variable constraints. Proper evaluation can confirm the solution set or the range of values that satisfy the inequality, paving the way for graphing solutions and understanding the system's behavior.