Problem 46 - Algebra 1

Question

Decide whether the ordered pair is a solution of the inequality. $$y>3 x^{2}+50 x+500 ;(-6,100)$$

Step-by-Step Solution

Answer

No, the ordered pair (-6,100) is not a solution of the inequality \(y > 3x^2 + 50x + 500\).

Step 1: Identification

Identify the ordered pair (-6,100) where the first value is x = -6 and the second value is y = 100.

Step 2: Substitution

Substitute these values into the inequality: 100 > 3(-6)^2 + 50(-6) + 500.

Step 3: Simplification

Simplify the right side of the inequality: 100 > 3*36 + (-300) + 500, which simplifies further to 100 > 108 - 300 + 500 and finally to 100 > 308.

Step 4: Comparison

Compare the left side (100) to the right side (308) and determine if the inequality stands: As 100 is not greater than 308, the provided ordered pair (-6,100) is not a solution to the inequality.

Key Concepts

Understanding Ordered Pairs

Ordered pairs are simply a set of two numbers written in a specific order. The first number refers to the value of \( x \), and the second number refers to the value of \( y \). For example, in the ordered pair \((-6, 100)\), \( x = -6 \) and \( y = 100 \). Ordered pairs are often used to represent points on a graph.
In context, they help us determine if specific values for \( x \) and \( y \) satisfy a given equation or inequality. By substituting these values into the equation, we can see if the ordered pair is a solution.

Solving Inequalities

Inequalities are mathematical statements that express a relationship of order between two expressions. In the inequality \( y > 3x^2 + 50x + 500 \), we are looking at whether \( y \) is greater than the expression on the right.
It's crucial to substitute the values of \( x \) and \( y \) from the ordered pair into the inequality to test if it holds true.

• If the statement is true, the ordered pair is a solution.
• If false, it's not a solution.

Understanding this process helps us find values that satisfy a range of conditions instead of just equalities.

Using the Substitution Method

The substitution method involves replacing variables with specific values to see how an equation or inequality behaves. Here, we substitute \( x = -6 \) and \( y = 100 \) into the inequality:
\[100 > 3(-6)^2 + 50(-6) + 500\]
This substitution helps us quickly assess if our ordered pair satisfies the inequality.
By working through the steps, we systematically determine if \( y \) is indeed greater than the calculated result on the right side. This method is a direct way to check solutions, especially with inequalities.

Simplification Process

The simplification process involves breaking down equations to make comparisons easier. After substituting values, we have:
\[100 > 3 \times 36 - 300 + 500\]
Simplify step-by-step to ensure clarity and accuracy:

• Perform multiplication: \( 3 \times 36 = 108 \)
• Add and subtract in sequence: \( 108 - 300 = -192 \) and \(-192 + 500 = 308 \)

Finally, compare \(100 > 308\). As 100 is not greater, the ordered pair is not a solution.
Simplification is key in solving mathematical problems as it helps verify the accuracy of potential solutions.