Problem 19 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ \frac{1-x}{4}<\frac{2 x-2}{3} $$

Step-by-Step Solution

Answer

The solution set is \( (1, \infty) \).

Step 1: Identify the Inequality

We need to solve \( \frac{1-x}{4} < \frac{2x-2}{3} \). This inequality has fractions on both sides.

Step 2: Eliminate the Fractions

To eliminate the fractions, find the least common multiple (LCM) of the denominators, which are 4 and 3. The LCM is 12. Multiply both sides of the inequality by 12 to clear the fractions: \[ 12 \cdot \frac{1-x}{4} < 12 \cdot \frac{2x-2}{3} \]This simplifies to: \[ 3(1-x) < 4(2x-2) \]

Step 3: Distribute the Factors

Distribute the factors in the inequality:\[ 3 - 3x < 8x - 8 \]

Step 4: Collect Like Terms

Move all terms involving \(x\) to one side and constant terms to the other side. Add \(3x\) to both sides and add \(8\) to both sides:\[ 3 + 8 < 8x + 3x \]This simplifies to:\[ 11 < 11x \]

Step 5: Solve for x

Divide both sides of the inequality by 11:\[ x > 1 \]

Step 6: Write the Solution in Interval Notation

The solution \(x > 1\) in interval notation is:\( (1, \infty) \)

Key Concepts

Inequality Solution

An inequality solution involves finding the range of values that satisfy a given inequality. Let's break it down with an example where you need to solve the inequality \( \frac{1-x}{4} < \frac{2x-2}{3} \).
To tackle this problem, you first identify and understand the components of the inequality. Notice that each side has a fraction. Therefore, a logical first step is to eliminate the fractions to make the inequality easier to solve.

• Find the Least Common Multiple (LCM) of the denominators, 4 and 3, which is 12.
• Multiply both sides of the inequality by the LCM to clear the fractions.
• This changes the inequality to \( 3(1-x) < 4(2x-2) \), simplifying your work.

Remember, at each transformation, the inequality should remain valid.

Interval Notation

Once you solve an inequality, expressing the solution in interval notation is key. This notation is a way to describe the range of values that satisfy the inequality in a concise format.
For example, if you solve \( 11 < 11x \) and find that \( x > 1 \), you describe the solution set using interval notation.

• Brackets or parentheses indicate the endpoints of the interval. A parenthesis \((\) means the endpoint is not included, whereas a bracket \([\) means it is included.
• The interval \((1, \infty)\) represents all numbers greater than 1, which solves \( x > 1 \).
  

Using interval notation helps communicate the solutions compactly and visually, making it easier to understand and interpret.

Algebraic Manipulation

Algebraic manipulation involves rearranging terms and using operations to solve for a variable. Let's see how this works in solving inequalities by looking at the inequality: \( 3(1-x) < 4(2x-2) \).
First, distribute the coefficients to eliminate parentheses:

• Expand: \( 3 - 3x < 8x - 8 \).
• Rearrange to bring all \(x\) terms to one side and constants to the other. Add \(3x\) and \(8\) to both sides: \( 3 + 8 < 11x \).
• This gives: \( 11 < 11x \).

Finally, solve for \(x\) by dividing both sides by 11, resulting in \(x > 1\).
Algebraic manipulation is a powerful tool for isolating variables and simplifying expressions, helping to arrive at a solution efficiently.