Problem 50 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The solution set is \( \{ x \mid x \leq \frac{4}{3} \} \).

Step 1: Expand Both Sides

Start by distributing the numbers outside the parentheses on both sides of the inequality. This means multiplying 3 with both terms inside the first set of parentheses and 4 with both terms inside the second set of parentheses.On the left side: \[ 3(5x - 4) = 15x - 12 \]On the right side: \[ 4(3x - 2) = 12x - 8 \]

Step 2: Set the Inequality

After expansion, rewrite the inequality with the expanded terms:\[ 15x - 12 \leq 12x - 8 \]

Step 3: Move Variable Terms to One Side

Subtract \(12x\) from both sides to get all the \(x\) terms on one side:\[ 15x - 12x - 12 \leq -8 \]This simplifies to:\[ 3x - 12 \leq -8 \]

Step 4: Isolate the Variable Term

Add 12 to both sides to isolate the term containing \(x\):\[ 3x - 12 + 12 \leq -8 + 12 \]This simplifies to:\[ 3x \leq 4 \]

Step 5: Solve for x

Divide both sides by 3 to solve for \(x\):\[ x \leq \frac{4}{3} \]

Step 6: Write the Solution in Solution Set Notation

Express the solution as a solution set:\[ \{ x \mid x \leq \frac{4}{3} \} \]

Key Concepts

Solution Set Notation

When you solve inequalities, finding a solution is just part of the process. Expressing it clearly is equally important. This is where solution set notation comes in handy. It allows us to represent the set of all possible solutions to an inequality. When you reach the last step of solving an inequality, and you have something like \( x \leq \frac{4}{3} \), you need to express it in solution set notation.

The notation is expressed as:

• Start with a curly brace: \( \{ \)
• State the variable: \( x \)
• Add a "bar" meaning "such that": \( \mid \)
• State the condition: the inequality itself, \( x \leq \frac{4}{3} \)
• Close the curly brace: \( \} \)

Therefore, \( \{ x \mid x \leq \frac{4}{3} \} \) reads as "the set of all \( x \), such that \( x \) is less than or equal to \( \frac{4}{3} \)." This form is neat and concise, making it easy to understand the range of your solution.

Distributive Property

The distributive property is a key algebraic principle used to simplify expressions and solve equations, particularly inequalities. It involves multiplying a single term by each term within a set of parentheses. In our inequality example, we used the distributive property at the very first step.

• Distribute the number outside the parentheses to each term inside.
• In our problem, distribute 3 over \( (5x - 4) \) to get \( 15x - 12 \).
• Similarly, distribute 4 over \( (3x - 2) \) to result in \( 12x - 8 \).

Keep in mind that the distributive property helps in breaking down complex expressions into simpler parts that are easier to manage. It is especially useful in inequalities as it allows you to directly work with each term. By applying the distributive property, the expression becomes more straightforward to handle, leading to a solution more efficiently.

Solving Linear Inequalities

Solving linear inequalities follows similar steps as solving linear equations but includes handling inequalities facts. The key part is maintaining the inequality throughout the process. In our given inequality, we started with \( 3(5x-4) \leq 4(3x-2) \), and here's how we solved it step-by-step:

• Expand using the distributive property: The first step was distributing, resulting in \( 15x - 12 \text{ and } 12x - 8 \).
• Rearrange: Bring all terms containing the variable \( x \) to one side by subtracting \( 12x \) from both sides, resulting in \( 3x - 12 \).
• Isolate the variable: Add 12 to both sides to get \( 3x \leq 4 \).
• Solve for \( x \): Divide each part by 3 which gives us \( x \leq \frac{4}{3} \).

Special care is necessary when multiplying or dividing both sides by a negative number, as this will flip the inequality sign. However, in this exercise, all multiplications and divisions were with positive numbers, so the inequality sign remained unchanged.