Problem 2 - Intermediate Algebra

Question

For Problems \(1-18\), solve each of the inequalities and express the solution sets in interval notation. $$ \frac{1}{4} x-\frac{4}{3} x<-13 $$

Step-by-Step Solution

Answer

\(x > 12\), or in interval notation, \((12, \infty)\).

Step 1: Identify and simplify the inequality

The inequality given is \( \frac{1}{4} x - \frac{4}{3} x < -13 \). First, combine the terms with \( x \).

Step 2: Find a common denominator

To combine the terms with \( x \), we need a common denominator for \( \frac{1}{4} \) and \( \frac{4}{3} \). The least common denominator is 12.

Step 3: Convert fractions to the common denominator

Convert \( \frac{1}{4} x \) to \( \frac{3}{12} x \) and \( \frac{4}{3} x \) to \( \frac{16}{12} x \). The inequality becomes \( \frac{3}{12} x - \frac{16}{12} x < -13 \).

Step 4: Simplify the inequality

Combine the fractions: \( \frac{3}{12} x - \frac{16}{12} x = \frac{-13}{12} x \). The inequality is now \( \frac{-13}{12} x < -13 \).

Step 5: Isolate \( x \)

To isolate \( x \), multiply both sides by the reciprocal of \( \frac{-13}{12} \), which is \( -\frac{12}{13} \). Note the inequality reverses direction because we multiply by a negative number. \( x > -13 \times -\frac{12}{13} \).

Step 6: Calculate the solution

Compute \( -13 \times -\frac{12}{13} = 12 \). Thus, the inequality simplifies to \( x > 12 \).

Step 7: Express the solution in interval notation

The solution \( x > 12 \) in interval notation is \((12, \infty)\).

Key Concepts

Interval Notation

Interval notation is a method of describing a set of numbers between two endpoints. This notation is particularly handy for representing solution sets of inequalities. Here's how it typically works:

• A round bracket, ( ), is used to express that an endpoint is not included in the set, known as an 'open' interval.
• A square bracket, [ ], indicates that an endpoint is included, called a 'closed' interval.

In interval notation, infinity (\( ∞ \) ) or negative infinity ( \(-∞ \) ) signifies that there is no bound on that side of the interval. Since infinity itself isn't an actual number, it's always combined with a round bracket.For example, the solution of our inequality, \( x > 12 \), means all numbers greater than 12 are included, but 12 itself is not. Thus, it's noted as (12, ∞ ).The elegance of interval notation lies in its simplicity. It effectively condenses the answer, making it easier to read and understand.

Combining Like Terms

Combining like terms in algebra helps to simplify expressions and solve equations or inequalities more efficiently. "Like terms" are those that have identical variable parts, meaning the variables must look exactly the same, including their exponents.In the inequality \(\frac{1}{4}x - \frac{4}{3}x < -13 \), both terms contain the variable \(x\). To combine them, you need a common denominator, which ensures that the fractions can be simplified accurately.After identifying 12 as the least common denominator for 4 and 3, you convert the terms:

• \( \frac{1}{4}x \) becomes \( \frac{3}{12}x \)
• \( - \frac{4}{3}x \) becomes \( -\frac{16}{12}x \)

Next, these are combined: \( \frac{3}{12}x - \frac{16}{12}x = -\frac{13}{12}x \).This simplification step reduces the burden of dealing with multiple terms, allowing you to focus on solving the inequality for \(x.\)

Fraction Manipulation

Fraction manipulation is a critical skill in algebra, particularly when solving equations and inequalities. It involves changing the form of fractions without altering their values to simplify calculations. When facing fractions with different denominators, the first step is to find the least common denominator (LCD) so that the fractions can be combined or compared directly.Consider the inequality \( \frac{1}{4}x - \frac{4}{3}x < -13 \). The denominators are 4 and 3, making the LCD 12. Once the fractions are converted to have this common denominator:

• \( \frac{1}{4}x \) is modified to \( \frac{3}{12}x \)
• \( -\frac{4}{3}x \) changes to \( -\frac{16}{12}x \)

This allows for straightforward arithmetic, combining the fractions into a single term: \( -\frac{13}{12}x \).Finally, multiplying both sides by the reciprocal, \(-\frac{12}{13}\), isolates \(x\), and flips the inequality due to the negative sign. This reversal is a pivotal moment, emphasizing the nuance of fraction management in inequality solutions.