Problem 55 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ \frac{1}{4}(x+4)<\frac{1}{5}(2 x+3) $$

Step-by-Step Solution

Answer

\( x > \frac{8}{3} \) or \( \{ x \mid x > \frac{8}{3} \} \).

Step 1: Eliminate Fractions

To eliminate the fractions, multiply both sides of the inequality by a common multiple of the denominators, which are 4 and 5. The least common multiple is 20. Thus, multiply each term by 20: \[ 20 \times \frac{1}{4}(x+4) < 20 \times \frac{1}{5}(2x+3) \] Simplifying gives: \[ 5(x+4) < 4(2x+3). \]

Step 2: Distribute the Multiplication

Distribute the numbers inside the parentheses: \[ 5 \times x + 5 \times 4 < 4 \times 2x + 4 \times 3 \] Simplifying gives: \[ 5x + 20 < 8x + 12.\]

Step 3: Rearrange Terms to Isolate Variable

To isolate the variable, first subtract \(5x\) from both sides of the inequality: \[ 20 < 3x + 12 \] Next, subtract 12 from both sides: \[ 8 < 3x.\]

Step 4: Solve for the Variable

To solve for \(x\), divide both sides by 3: \[ \frac{8}{3} < x \] This can also be written as: \[ x > \frac{8}{3}. \]

Step 5: Write Solution in Set Notation

Express the solution from the previous step in set notation. The solution set for the inequality is: \( \{ x \mid x > \frac{8}{3} \} \).

Key Concepts

Solution Set Notation

When you solve inequalities, expressing your answer in the correct format is important. Solution set notation helps articulate the set of numbers that satisfy an inequality. Think of it like a clothesline to "hang" your solutions neatly.

This notation uses a curly bracket to enclose the solutions. Inside the bracket, write the variable and the condition it must satisfy. For instance:

• \( \{ x \mid x > \frac{8}{3} \} \) means any value of \( x \) that is greater than \( \frac{8}{3} \)

Here, the vertical bar "\( \mid \)" reads as "such that," connecting the variable with its condition. This notation ensures that whoever reads it knows exactly which numbers can be used as solutions.

Solving Inequalities

Inequalities are like equations but they show a range of possible solutions. Think of an inequality as a scale, which can tilt in different directions. They use symbols like "<" (less than), ">" (greater than), "≤" (less than or equal to), and "≥" (greater than or equal to).

To solve an inequality:

• Treat them like equations, performing operations on both sides.
• Be cautious when dividing or multiplying by a negative number, as this flips the inequality sign.

In this exercise, you learned to deal with inequalities and how operations could affect them. When you multiplied the entire inequality by 20, you removed fractions without changing their direction. This makes them easier to manage and solve.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions or equations. In solving inequalities, it involves crossing hurdles like fractions to bring terms to one side and variables to the other.

Key strategies include:

• Eliminating fractions: Multiply by the least common multiple of denominators to simplify.
• Distributing terms: Multiply each term inside parentheses by the factor outside.
• Isolating the variable: Move terms with variables to one side, constants to the other.

In the exercise, you distributed and simplified the terms, making the inequality straightforward to solve. Rearranging terms skillfully allowed you to isolate \( x \) and find its range as shown in your solution set notation, ensuring clarity and precision in your final answer.