Problem 1 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

(a) '<', (b) '≤', (c) '≤', (d) '>'.

Step 1: Analyze Expression (a)

Given inequality is \( x < 5 \). Subtract 3 from both sides of the inequality: \( x - 3 < 5 - 3 \). This simplifies to \( x - 3 < 2 \). Therefore, the inequality sign that fills the blank is '<'.

Step 2: Solve Expression (b)

The given inequality is \( x \leq 5 \). Multiply both sides by 3: \( 3x \leq 3 \times 5 \). This results in \( 3x \leq 15 \). Therefore, the inequality sign for this blank is '≤'.

Step 3: Evaluate Expression (c)

Here, the inequality is \( x \geq 2 \). Multiply both sides by -3, remembering to flip the inequality sign: \(-3x \leq -3 \times 2 \). This simplifies to \(-3x \leq -6 \). Hence, the inequality that fits is '≤'.

Step 4: Process Expression (d)

Given inequality is \( x < -2 \). Multiply both sides by -1, flipping the inequality sign: \( -x > -(-2) \). This becomes \( -x > 2 \). Hence, the correct inequality sign is '>'.

Key Concepts

Inequality Signs

Inequality signs are symbols used to compare two values or expressions. They show how one quantity relates to another. The most common inequality signs are:

• <: meaning "less than"
• ≤: meaning "less than or equal to"
• >: meaning "greater than"
• ≥: meaning "greater than or equal to"

Read more about inequality signs here.
When solving inequalities, it's crucial to pay attention to these signs as they determine the relationship between quantities. For example, in the inequality \( x < 5 \), it indicates that \( x \) can be any number less than 5.
Similarly, knowing how to manipulate these signs, especially during multiplication or division by a negative number, is essential. This involves reversing or "flipping" the inequality sign. Understanding these symbols helps greatly in solving and interpreting inequalities correctly.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like plus or minus). They allow us to describe general mathematical ideas and relationships.
In our contexts, expressions are used in inequalities to compare two different expressions. For example, in \( x - 3 < 2 \), \( x - 3 \) is an algebraic expression, and it represents any value less than 2.

• Variables: symbols that represent unknown values, like \( x \) or \( y \)
• Operators: symbols that represent operations, such as \( +, -, \, \times, \div \)

Understanding how to manipulate and rearrange these expressions is foundational in algebra. This includes combining like terms or using operations like addition or multiplication to isolate variables when solving inequalities.

Solving Inequalities

To solve inequalities effectively, one must apply similar methods used in solving equations, but with special rules for inequality signs.
Here are key steps:

• Isolate the variable on one side through addition, subtraction, multiplication, or division.
• If multiplying or dividing by a negative number, flip the inequality sign.
• Simplify the expressions as needed without changing the inequality's direction, unless required.

For instance, in the problem \( -3x \leq -6 \), multiplying both sides by \(-1\) necessitates flipping the "less than or equal to" sign to "greater than or equal to." This is a unique aspect of inequalities that doesn't occur with standard equations. Mastering these techniques allows you to solve for the variable's possible values accurately.

Mathematical Reasoning

Mathematical reasoning refers to the logical thought process used to solve problems, understand concepts, and draw conclusions. It's the backbone of solving inequalities as it guides you through analyzing and manipulating expressions.
When approaching an inequality problem, it's vital to consider:

• What the inequality indicates about the values of the variable.
• How operations affect both sides of the inequality.
• Logical steps needed to isolate the variable or interpret the result.

For example, analyzing \( x < -2 \) in our exercise requires reasoning that if \(-x\) is the subject, multiplying by \(-1\) will flip the inequality to \(-x > 2\). This change reflects how the manipulation impacts the overall inequality.
Using sound reasoning ensures clarity and accuracy in both solving and concluding the steps of inequalities.