Problem 49 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ 4(3 x-1) \leq 5(2 x-4) $$

Step-by-Step Solution

Answer

\(x \leq -8\); solution set: \(\{x \mid x \leq -8\}\)

Step 1: Expand Both Sides

Expand both sides of the inequality by distributing the numbers outside the parentheses. On the left-hand side: \[4(3x - 1) = 12x - 4\] On the right-hand side: \[5(2x - 4) = 10x - 20\]The inequality becomes: \[12x - 4 \leq 10x - 20\]

Step 2: Move Variable Terms to One Side

Subtract \(10x\) from both sides to get all the variable terms on one side:\[12x - 10x - 4 \leq -20\] which simplifies to:\[2x - 4 \leq -20\]

Step 3: Isolate the Variable

Add 4 to both sides to isolate the term with the variable:\[2x - 4 + 4 \leq -20 + 4\] This simplifies to:\[2x \leq -16\]

Step 4: Solve for x

Divide both sides by 2 to solve for \(x\):\[\frac{2x}{2} \leq \frac{-16}{2}\] Which simplifies to:\[x \leq -8\]

Step 5: Write the Solution in Set Notation

Express the solution in set notation:\[\{x \mid x \leq -8\}\]

Key Concepts

Solution Set Notation

When solving inequalities, one of the key tasks is to express the solution in a clear and precise way. Solution set notation is perfect for this task. Instead of just giving a single number, we describe all possible solutions.
A solution set is usually written using curly braces. For example, the solution set for the inequality \(x \leq -8\) is written as \( \{x \mid x \leq -8\} \).

• The curly braces \(\{ \} \) indicate a set of numbers.
• The vertical bar \(\mid\) means "such that."
• It shows that every number less than or equal to -8 is part of the solution.

In this context, the solution set notation is a simple tool to capture all possible solutions in one go, ensuring nothing is left out.

Distributive Property

The distributive property is an essential algebraic rule used to simplify expressions. It's the rule that lets us "distribute" a factor across terms inside a parenthesis.
In the equation \(4(3x - 1) \leq 5(2x - 4)\), the distributive property allows us to expand both sides:

• For the left side: \(4 \times 3x - 4 \times 1 = 12x - 4\).
• For the right side: \(5 \times 2x - 5 \times 4 = 10x - 20\).

This step is crucial for breaking down more complex expressions into simpler terms.
Once expanded, the inequality becomes \(12x - 4 \leq 10x - 20\), ready for further simplification.

Isolating Variables

To solve inequalities, isolating the variable is a fundamental process. The aim is to get the variable by itself on one side of the inequality.
Here are the steps used to isolate the variable \(x\):

• First, subtract \(10x\) from both sides: \(12x - 10x - 4 \leq -20\). This results in \(2x - 4 \leq -20\).
• Next, add 4 to both sides to further isolate \(2x\): \(2x \leq -16\).
• Finally, divide both sides by 2 to get \(x\) alone: \(x \leq -8\).

Each mathematical operation maintains the balance of the inequality. The trick is to perform the same operation on both sides, revealing the true scope of the variable's limits.

Set Notation in Inequalities

Set notation in inequalities is a precise way to capture all solutions quickly and efficiently. When we have an inequality like \(x \leq -8\), set notation allows us to present this information compactly.The notation \( \{x \mid x \leq -8\} \) tells us exactly what the solutions to this inequality are. Here's how it works:

• "\(x\)" begins the description of possible values.
• "\(\mid\)" means "such that."
• "\(x \leq -8\)" defines that \(x\) can be any number less than or equal to -8.

Using set notation helps to avoid confusion by clearly specifying all the numbers that satisfy the inequality. This method is an integral part of communicating mathematical solutions effectively.