Problem 62 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The ordered pair \((0,-3)\) satisfies \(y>2 x-3\)

Step-by-Step Solution

Answer

The statement is false. The correct inequality is \(y \leq 2x - 3\).

Step 1: Substitute the ordered pair into the inequality

Replace x with 0 and y with -3 in the inequality \(y > 2x - 3\). This results in \(-3 > 2(0) - 3\) or \(-3 > -3\).

Step 2: Evaluate the inequality

Simplify \(-3 > -3\) to check if it is true. The simplified expression is false because -3 is not greater than -3.

Step 3: Make necessary changes to produce a true statement

The original inequality doesn't hold true for the given ordered pair (0,-3). To correct the mismatch, the inequality symbol needs to be changed from '>' (greater than) to '\leq' (less than or equal to). Our new and corrected inequality becomes \(y \leq 2x - 3\). When we substitute (0,-3) back into the corrected inequality, we get \(-3 \leq -3\) which is a true statement.

Key Concepts

Ordered Pairs

In mathematics, ordered pairs are quite simple yet incredibly important. An ordered pair is a pair of numbers written in a specific order, typically as \(x, y\). Here, \(x\) is the first element, and \(y\) is the second element. The position of each number is crucial because it defines its role.
Ordered pairs are commonly used to describe points in a plane.
For example, the ordered pair \(0, -3\) represents a specific point on the coordinate plane where \(x = 0\) and \(y = -3\). This tells us that the point is located at zero on the x-axis and negative three on the y-axis.
Ordered pairs are used in various mathematical contexts such as functions, graphs, and geometric shapes.

Substitution Method

To check if an ordered pair satisfies a given inequality, we use the substitution method. This involves replacing the variables in the inequality with the values from the ordered pair.
Take the example inequality \(y > 2x - 3\), and the ordered pair \(0, -3\). Here, we substitute \(x = 0\) and \(y = -3\) into the inequality.
This results in \(-3 > 2(0) - 3\),
which further simplifies to \(-3 > -3\).

• This process helps determine whether the statement holds true for the given values.
• If the inequality is satisfied, the statement is true.
• If it's not, as in our example, the inequality needs revision to correctly represent the condition that's true for the pair.

In problems like this, use substitution to test and verify solutions.

Inequality Symbols

Inequality symbols are used to compare two values or expressions. Common inequality symbols include:

• \(>\) (greater than)
• \(<\) (less than)
• \(\geq\) (greater than or equal to)
• \(\leq\) (less than or equal to)

Understanding these symbols is vital for solving inequalities correctly, as in mathematics, the meaning can be completely altered by a small change. For example, the symbol \(>\) means the value on the left is greater than the one on the right,
while \(\leq\) allows the left side to be equal to or lesser than the right side.
In the initial problem, we saw that the inequality \(y > 2x - 3\) did not hold for the ordered pair \(0, -3\) because the expression \(-3 > -3\) is false.
When modifying the inequality symbol to \(\leq\), the new inequality \(y \leq 2x - 3\) correctly represented the condition \(-3 \leq -3\), making it true.
Thus, knowing the correct application of these symbols is crucial in forming true mathematical statements.