Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Copy and complete the statement using the correct inequality symbol. If \(3 x>9\), then \(x\) \(?_____ 3 .\)

Step-by-Step Solution

Answer

The completed statement is: If \(3x > 9\), then \(x > 3\).

Step 1: Analyse and set up the given inequality

Firstly, we are supposed to compare 'x' to the number '3' using the inequality '3x > 9'. Our goal will be to isolate 'x' on one side of the inequality.

Step 2: Solve the inequality for 'x'

To isolate 'x', the term '3x' is divided by '3'. The inequality becomes '3x/3 > 9/3', which simplifies to 'x > 3'.

Step 3: Comparing 'x' and '3'

Now, it is clear that 'x' is greater than '3'. Therefore, the correct inequality symbol to use to complete the statement is '>'.

Key Concepts

Linear Inequalities

Linear inequalities are similar to linear equations, but instead of having an equal sign, they have one of the inequality symbols. They express a relationship where one value is not exactly equal to another but is greater than, less than, or perhaps greater than or equal to, or less than or equal to. Just like linear equations, linear inequalities can involve one variable, and they form a straight line if graphed in a coordinate plane with two variables.
Linear inequalities can take many forms. Here are some examples:

• \(x + 3 > 5\)
• \(2y - 1 \leq 4\)

When dealing with linear inequalities, the solution set includes a range of values that satisfy the inequality. For example, in the inequality \(x > 3\), any number greater than 3 satisfies the inequality.

Solving Inequalities

Solving inequalities follows a process very similar to solving equations, with a few extra rules to keep in mind. The goal is to isolate the variable on one side of the inequality to find the set of solutions. Here’s a simple guideline:
1. **Isolate the Variable**: Start by moving all terms involving the variable to one side, and the constant terms to the other. Just like equations, you can add, subtract, multiply, and divide both sides by a number to do this.
2. **Reverse the Inequality**: One very important rule is that, if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. For instance, if you start with \(-2x < 6\) and divide both sides by -2, the inequality sign changes to \(x > -3\).
3. **Simplify**: Finally, simplify the inequality to find the solution. For instance, in the problem \(3x > 9\), dividing both sides by 3 gives \(x > 3\). This means any number greater than 3 is a solution to the inequality.

Inequality Symbols

Inequality symbols are used to compare two expressions. They play a crucial role in expressing inequalities. Here's a quick rundown of the common inequality symbols and what they mean:

• **Greater than (\(>\))**: This symbol indicates that the expression on the left of the symbol is larger than the expression on the right. For example, \(x > 3\) means that \(x\) is greater than 3.
• **Less than (\(<\))**: This indicates that the expression on the left is smaller. For example, \(x < 5\) means \(x\) is less than 5.
• **Greater than or equal to (\(\geq\))**: This symbol means the expression is either greater than or exactly equal to another.
• **Less than or equal to (\(\leq\))**: This serves the opposite function, indicating the expression is less than or equal to another.

Understanding these symbols is key to correctly interpreting and solving problems involving inequalities.