Problem 12 - Intermediate Algebra

Question

For Problems \(1-18\), solve each of the inequalities and express the solution sets in interval notation. $$ \frac{3 x+2}{9}-\frac{2 x+1}{3}>-1 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, \frac{8}{3})\).

Step 1: Eliminate Fractions by Finding a Common Denominator

To simplify the inequality, identify the least common denominator (LCD) of the fractions. Here, the denominators are 9 and 3, and the LCD is 9. Multiply each term by 9 to clear the fractions:\[9 \left( \frac{3x+2}{9} \right) - 9 \left( \frac{2x+1}{3} \right) > 9(-1) \]Simplifying gives:\[3x + 2 - 3(2x + 1) > -9\]

Step 2: Distribute and Combine Like Terms

Distribute the multiplication across the terms:\[3x + 2 - (6x + 3) > -9\]Combine like terms:\[3x + 2 - 6x - 3 > -9\]This simplifies to:\[-3x - 1 > -9\]

Step 3: Isolate the Variable Term

Add 1 to both sides to isolate the term with the variable:\[-3x - 1 + 1 > -9 + 1\]This simplifies to:\[-3x > -8\]

Step 4: Solve for the Variable

Divide both sides by -3 to solve for \(x\). Remember, when dividing by a negative number, the inequality sign flips:\[x < \frac{8}{3}\]

Step 5: Express the Solution in Interval Notation

The solution in interval notation is expressed as all numbers less than \(\frac{8}{3}\):\[(-\infty, \frac{8}{3})\]

Key Concepts

Interval Notation

Interval notation is a way of representing a range of numbers along a number line. It is highly efficient and helps quickly identify where solutions for inequalities begin and end. In the interval notation, parentheses \((\) or \()\) indicate that a boundary point is not included (open interval), while square brackets \([\) or \()]\) suggest that it's included (closed interval).
For instance, if we solve an inequality and find that the solution set is all values less than a certain number, such as \(x < \frac{8}{3}\), we express this range in interval notation as \((-\infty, \frac{8}{3})\). The open parenthesis with infinity implies that \(-\infty\) is not a boundary, as infinity itself is a concept, not a number.
Using the open parenthesis with \(\frac{8}{3}\) tells us \(\frac{8}{3}\) is not part of the solution, aligning with \(x < \frac{8}{3}\). This efficient form of notation simplifies the representation of solutions, saving space and time.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. Finding a common denominator is essential when you want to add, subtract, or compare fractions. It simplifies the process, especially when dealing with equations or inequalities involving fractions.
In the given inequality problem, we had two fractions: \(\frac{3x+2}{9}\) and \(\frac{2x+1}{3}\). The denominators are 9 and 3, and their least common denominator (LCD) is 9. This is because 9 is the smallest number into which both denominators divide evenly.
After identifying the LCD, multiply every term in the inequality by it to eliminate the fractions. This results in:

• \(9 \left( \frac{3x+2}{9} \right) \) simplifies to \(3x + 2\)
• \(9 \left( \frac{2x+1}{3} \right) \) becomes \(3(2x + 1)\)

This technique clears fractions, simplifying the task of solving inequalities.

Combining Like Terms

Once you have an equation or inequality free of fractions, the next step is combining like terms. Like terms are terms that have the same variable raised to the same power. Simplifying expressions by combining like terms makes solving inequalities or equations more manageable.
In the step-by-step solution, after eliminating the fractions in our inequality:\[3x + 2 - (6x + 3) > -9\]You distribute and simplify:

• \(3x + 2\) are the terms from the first part.
• Subtract \(6x + 3\), resulting from multiplying \((2x + 1)\) by 3.

Combining like terms yields:

• Take \(3x\) and subtract \(6x\) to get \(-3x\).
• Subtract \(3\) from \(2\), leaving \(-1\).

These steps result in the simplified expression \(-3x - 1 > -9\), which is much easier to work with.

Solving Inequalities

Solving inequalities is similar to solving equations but with additional rules due to the inequality signs involved. When working on inequalities, keep in mind that the sign indicates a range of possible solutions and changes under specific operations.
Here, after simplifying the inequality to \(-3x - 1 > -9\):

• We isolate the variable by moving constants to the other side, producing \(-3x > -8\).
• The next step divides both sides by \(-3\). Remember, dividing or multiplying an inequality by a negative number flips the inequality sign. So, \(x < \frac{8}{3}\).

This is a crucial difference from solving equations and underscores how inequalities account for a broader range of solutions. The solution describes all values of \(x\) that satisfy the inequality, demonstrated clearly using interval notation. Solving inequalities demands careful attention to these rules to ensure accurate solutions.