Problem 12 - Algebra and Trigonometry

Question

Systems of Equations and Inequalities. $$y>-3$$

Step-by-Step Solution

Answer

The solution is all real numbers greater than -3.

Step 1: Interpret the inequality

The inequality \(y > -3\) represents all points in the coordinate plane where the \(y\)-coordinate is greater than \(-3\).

Step 2: Describe the solution

This is the half-plane above the horizontal line \(y = -3\). The boundary line \(y = -3\) is a dashed line (since the inequality is strict, not \(\geq\)). The solution region is all points above this line.

Key Concepts

inequalities

Inequalities involve mathematical expressions that show the relationship between two values or expressions. Instead of showing a strict equivalence like equations, inequalities depict a range of values. Let's take the inequality \( y > -3 \) as an example.

In this context, the inequality indicates which numbers the variable \( y \) can take. Often, such inequalities are based on the comparison of two quantities, demonstrating that one value is larger or smaller than the other, but not always equal. Here are some key points about inequalities:

• They include symbols like \(<\), \(>\), \(\leq\), and \(\geq\).
• Inequalities do not have a unique solution but rather a solution set.
• Compound inequalities can express more complex relationships, such as \(-4 < x \leq 2\).

For the case of \( y > -3 \), it means values of \( y \) stretch from just above -3 to positive infinity, emphasizing that even though -2 is part of the solution set, precisely -3 is not because the inequality does not include the equality symbol (\(=\)).

real numbers

Real numbers are all the numbers that exist on the number line. They encompass both rational numbers, like fractions and integers, as well as irrational numbers, like \(\pi\) and \(\sqrt{2}\). Real numbers form the continuum in mathematics, representing virtually all feasible quantities.

The landscape of real numbers can be broken down as follows:

• Integers: Whole numbers and their negatives, including zero (e.g., -3, 0, 5).
• Rational numbers: Numbers expressible as a fraction of two integers, with a non-zero denominator (e.g., \(3/4\), \(-2/1\)).
• Irrational numbers: Numbers that cannot be expressed as fractions, with non-terminating, non-repeating decimals (e.g., \(\pi\), \(e\)).
• Whole and Natural numbers: A subset of integers without negative numbers (e.g., 0, 1, 2).

Understanding that \( y > -3 \) encompasses all real numbers greater than -3 highlights the infinite range of numbers involved, stressing the importance of real numbers in expressing expansive solutions like those seen in inequalities.

greater than symbol

The greater than symbol (>) is a key operator in mathematics used to indicate that one quantity is larger than another. Its introduction marks one of the building blocks for understanding mathematical relationships beyond simple equality.

In the inequality \( y > -3 \), the symbol '>' delineates that \( y \) is not just any number, but specifically one that exceeds -3. Here's what you need to know about the greater than symbol:

• It makes an unambiguous distinction between numbers: wherever '>' is used, it implies the number on the left is larger.
• The inequality \( y > -3 \) does not include \( -3 \) itself in its solution. In mathematical terms, the boundary (\(-3\)) is not part of the solution set.
• When using '>' symmetrical with '<', it allows comparison both ways, elucidating the inverse relationship (e.g., if \( a > b \), then \( b < a \)).

This understanding aids in solving, graphing, and interpreting inequalities, providing clarity in solving systems involving inequalities.