Problem 91 - Introductory Algebra for College Students

Question

Use properties of inequality to rewrite each inequality so that \(x\) is isolated on one side. \(3 x+a>b\)

Step-by-Step Solution

Answer

\(x>\frac{b-a}{3}\)

Step 1: Identify the variable term to be isolated

In the given inequality \(3x+a>b\), the variable to be isolated is \(x\). Here, the variable term is '3x'.

Step 2: Subtract 'a' from both sides

In order to isolate \(x\), subtract \(a\) from both sides of the inequality, that will give you a new inequality: \(3x+a-a>b-a\). After subtracting you get \(3x>b-a\).

Step 3: Divide all terms by 3

Given that the variable term is \(3x\), to isolate \(x\), you divide all terms by 3. That gives you: \(x>\frac{b-a}{3}\).

Key Concepts

Properties of Inequality

In algebra, inequalities are expressions that show how two values relate to each other when they are not necessarily equal. Understanding the properties of inequalities is key to manipulating and solving them. The properties that apply to inequalities are similar to those used in equations, but with a few extra considerations.

• **Transitive Property**: If a > b and b > c, then a > c.
• **Addition/Subtraction Property**: You can add or subtract the same number from both sides of an inequality without changing its direction. For example, if a > b, then a + c > b + c.
• **Multiplication/Division Property**: If you multiply or divide both sides of an inequality by a positive number, the inequality remains the same. For example, if a > b, then ac > bc for c > 0. However, if you multiply or divide by a negative number, the inequality sign flips! If a > b, then ac < bc for c < 0.

These properties allow you to manipulate inequalities, similar to how you solve equations, but always remember the specific rule about changing the inequality sign when dealing with negative numbers.

Isolating Variable

Isolating the variable means getting the variable alone on one side of the inequality. This is a crucial step in solving inequalities, just like in solving equations. When you isolate a variable, you make it easier to understand the relationship between the variable and the other numbers.
To isolate a variable like our example with the inequality \(3x + a > b\):

• **Identify the Variable**: In \(3x + a > b\), the term with the variable is \(3x\).
• **Move Other Terms**: We first focus on moving the term \(a\) away from \(3x\), by subtracting \(a\) from both sides of the inequality.
• **Perform Operations**: Once \(a\) is subtracted, simplify to get \(3x > b - a\).

The outcome is that the variable \(x\) is closer to being isolated. You continue to perform operations to make the variable stand alone.

Solving Inequalities

Solving inequalities involves finding the set of all possible values of the variable that make the inequality true. Much like solving equations, this process requires systematically applying the properties of inequality.
For example, once you have isolated your variable term in an inequality like \(3x > b - a\), you continue:

• **Dividing to Isolate**: To solve for \(x\), you need to separate it completely. In this case, divide every part of the inequality by 3, giving us \(x > \frac{b-a}{3}\).
• **Checking Your Solution**: After solving, it’s important to ensure your solution satisfies the original inequality. Substitute numbers back to check.
• **Interpreting the Result**: The solution \(x > \frac{b-a}{3}\) shows all values greater than \(\frac{b-a}{3}\) are solutions to the inequality. This understanding is crucial for graphs or further applications.

Remember, solving inequalities is about finding ranges of numbers, not just one number. Practice with different inequalities helps to solidify the method and build intuition for these expressions.