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Chapter 3: Problem 8

Is the ordered pair a solution to the given inequality? $$ 7 x-3 y<21 ;(5,-3) $$

Short Answer

Expert verified
The ordered pair (5, -3) is not a solution to the inequality, because 44 is not less than 21.

Step by step solution

01

Understand the Inequality

The inequality given is \( 7x - 3y < 21 \). We need to determine if the ordered pair \((5, -3)\) is a solution. This means we substitute \(x = 5\) and \(y = -3\) into the inequality to see if it holds true.
02

Substitute the Ordered Pair

Substitute \(x = 5\) and \(y = -3\) into the inequality. This gives us: \[ 7(5) - 3(-3) < 21 \] Calculate the left-hand side to check if the inequality is satisfied.
03

Calculate Left-hand Side

Perform the calculations: \[ 7(5) = 35 \]\[ -3(-3) = 9 \] Substitute these values back into the inequality: \[ 35 + 9 < 21 \]
04

Simplify and Check the Inequality

Simplify the expression: \[ 44 < 21 \]This statement is false. Therefore, the inequality does not hold true for the given ordered pair \((5, -3)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Ordered Pairs
When working with inequalities, an ordered pair often helps us determine whether a point lies within the solution set of the inequality. An ordered pair consists of two elements, usually written in the form \( (x, y) \).
In our exercise, the ordered pair \( (5, -3) \) represents specific values of\( x \) and \( y \) that are tested against the given inequality.
The \( x \) value corresponds to the horizontal coordinate, and the \( y \)value corresponds to the vertical coordinate on a graph.
To determine if an ordered pair satisfies an inequality, we substitute the \( x \) and \( y \)values into the inequality and check if the resulting statement is true.
The Art of Inequality Substitution
Inequality substitution is the process where we substitute specific values of the variables into an inequality to evaluate its truthfulness.
For our given inequality \( 7x - 3y < 21 \), we substitute \( x = 5 \) and\( y = -3 \).
Once substituted, the inequality becomes:
  • Calculate \( 7(5) \) which equals \( 35 \).
  • Calculate \( -3(-3) \) which results in \( 9 \).
The substituted inequality then reads \( 35 + 9 < 21 \).
This step is crucial because it transforms an abstract mathematical expression into a numerical comparison, allowing for a clear verification of the inequality.
Inequality Verification and Checking
Verification is the final step in determining if a particular ordered pair is a solution to an inequality. After substitution, our inequality is \( 44 < 21 \).
This is a straightforward numerical check. If the inequality holds true, meaning the left-hand side is indeed less than the right-hand side, the ordered pair is a solution.
In our example, \( 44 < 21 \) is false. Therefore, \( (5, -3) \) is not a solution to \( 7x - 3y < 21 \).
Checking whether an inequality holds helps confirm which parts of a coordinate plane satisfy the inequality conditions. This understanding is vital for solving more complex systems involving multiple inequalities.

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