Problem 11 - Algebra for College Students

Question

Solve each of the inequalities and express the solution sets in interval notation. \(\frac{4 x-3}{6}-\frac{2 x-1}{12}<-2\)

Step-by-Step Solution

Answer

The solution is \((-\infty, -\frac{19}{6}).\)

Step 1: Simplify the Left Side

The goal is to have one fraction on the left side. We'll find a common denominator for the fractions. Notice that 6 and 12 are both divisible by 12. Multiply the first fraction by 2 to have a common denominator of 12: \[\frac{4x-3}{6} \text{ becomes } \frac{2(4x-3)}{12} = \frac{8x-6}{12}.\] Now we can rewrite and combine the fractions: \[\frac{8x-6}{12} - \frac{2x-1}{12} = \frac{8x-6-(2x-1)}{12} = \frac{6x-5}{12}.\]

Step 2: Solve the Inequality

We'll clear the fraction by multiplying both sides of the inequality by 12, making the inequality:\[6x-5 < -24.\]Next, add 5 to both sides:\[6x < -19.\]Finally, divide both sides by 6:\[x < -\frac{19}{6}.\]

Step 3: Express the Solution in Interval Notation

The inequality \(x < -\frac{19}{6}\) means that \(x\) can take any value less than \(-\frac{19}{6}\). In interval notation, this is expressed as:\[(-\infty, -\frac{19}{6}).\]

Key Concepts

Fraction Simplification

Fraction simplification is an essential skill in algebra. It helps make calculations easier and clearer. When you have complex fractions, the goal is to combine them into one fraction with a single denominator.
This process often starts with finding a common denominator. In the exercise, the fractions are \( \frac{4x-3}{6} \) and \( \frac{2x-1}{12} \). Their common denominator is 12.

• First, adjust the fraction \( \frac{4x-3}{6} \) so it has 12 as a denominator. Multiply both the numerator and denominator by 2 to get \( \frac{8x-6}{12} \).
• This step does not change the fraction's value, just its appearance, making it easier to combine with the other fraction.

Once both fractions have the same denominators, subtract their numerators:
\( \frac{8x-6}{12} - \frac{2x-1}{12} = \frac{(8x-6) - (2x-1)}{12} \).

Simplify the numerator to get \( \frac{6x - 5}{12} \). This is a key step in solving equations with fractions.

Interval Notation

Interval notation is a concise way to express a range of numbers. It's frequently used to show solutions of inequalities. With interval notation, you can easily see which numbers are included or excluded in the solution.

• Here, we use parentheses \(()\) for not including endpoints. If you need to include the endpoints, brackets \([]\) are used instead.

In the exercise, the solution to the inequality \(x < -\frac{19}{6}\) is expressed in interval notation as \((-\infty, -\frac{19}{6})\).
This notation shows that all numbers less than \(-\frac{19}{6}\) are included in the solution, but not \(-\frac{19}{6}\) itself.

By using this notation, we provide a clear, compact representation of possible solutions.

Learn to appreciate it as it simplifies the presentation of solutions immensely.

Inequality Solution

Solving inequalities involves finding all possible values that satisfy the condition. It's similar to solving regular equations, but there are key differences to keep in mind.
In the given exercise, after simplifying the fraction, we have the inequality:

• First, \(6x-5 < -24\). We eliminate the fraction by multiplying both sides by 12.
• Adding 5 to both sides, we have \(6x < -19\).
• Finally, dividing both by 6 gives \(x < -\frac{19}{6}\).

This systematic approach involves:

• Performing operations equally on both sides to maintain balance.
• Ensuring that when multiplying/dividing both sides by a negative number, the inequality sign reverses. Though not directly relevant here, it’s crucial in other contexts.

The final solution \(x < -\frac{19}{6}\) means any number less than \(-\frac{19}{6}\) satisfies the inequality, marking this as complete.