Problem 1 - Algebra and Trigonometry

Question

Fill in the blank with an appropriate inequality sign. (a) If \(x<5,\) then \(x-3 ________\) (b) If \(x \leq 5,\) then 3\(x ________ 15\) (c) If \(x \geq 2,\) then \(-3x _______ -6\) (d) If \(x<-2,\) then \(-x _________ 2\)

Step-by-Step Solution

Answer

(a) <, (b) \leq, (c) \leq, (d) >

Step 1: Analyze (a)

We start with the inequality \(x < 5\). To find \(x - 3\), subtract 3 from both sides: \(x - 3 < 5 - 3\). Thus, \(x - 3 < 2\).

Step 2: Analyze (b)

We have the inequality \(x \leq 5\). To find 3\(x\), multiply all terms by 3: \(3x \leq 3 \times 5\). Therefore, 3\(x \leq 15\).

Step 3: Analyze (c)

Given \(x \geq 2\), multiply all terms by -3 and reverse the inequality because of multiplying by a negative: \(-3x \leq -3 \times 2\). This gives us \(-3x \leq -6\).

Step 4: Analyze (d)

Starting with \(x < -2\), multiply all sides by -1 and reverse the inequality: \(-x > -1 \times -2\). Hence, \(-x > 2\).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions. It helps us describe problems and perform operations in a straightforward manner.
In algebra, we frequently encounter expressions such as \(x-3\), \(3x\), and \(-3x\). These expressions represent various mathematical relationships or operations involving the variable \(x\).
When solving algebraic problems, we manipulate these expressions to simplify or solve equations. Understanding how to manipulate letters and symbols is key. It allows us to transfer real-world problems into mathematical language. By doing so, we can solve these problems using known mathematical techniques.

Inequality Signs

Inequality signs are tools that help us compare two values. Common inequality symbols include:

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

Inequalities are like equations, but they do not have equal outcomes. They express a relationship of size, where one quantity is either smaller than, greater than, or not equal to another.
When working with inequalities, be aware of operations involving negative numbers. Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.
Correct usage of inequality symbols allows us to frame solutions and explore different possibilities within a mathematical problem.

Problem-Solving

Problem-solving in mathematics involves identifying a problem, developing a strategy, and applying known methods to find an answer. This process is central to understanding and effectively using inequalities.
To solve an inequality problem, follow these steps:

• Identify the given condition, such as \(x < 5\) or \(x \geq 2\).
• Determine what mathematical operation or transformation needs to be conducted.
• Apply the operation carefully, maintaining the integrity of the inequality.
• Evaluate whether special rules apply, such as reversing the inequality sign when multiplying by a negative number.

By practicing this structured approach, you’ll gain confidence in solving a broad range of inequality problems. It helps you look at problems methodically and reach logical solutions.

Mathematical Reasoning

Mathematical reasoning is the logical thought process used to understand, explore, and solve mathematical problems. It is essential in solving inequalities effectively.
In mathematical reasoning, one starts with a given set of conditions or inequalities. Using rules of algebra and logic, you develop a sequence of steps to reach a conclusion. For instance, multiplying both sides of an inequality by 3 requires checking if the operation affects the inequality sign.
Here’s how reasoning helps resolve each given inequality:

• For example (a) and (b), focusing on balances and operations helps maintain the original relationship.
• In examples (c) and (d), understanding the impact of negative multipliers is critical.

Reasoning ability is developed through practice and application. As a skill, it helps you not just in math, but in problem-solving across various domains.