Problem 21 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{1}{3} x+2<\frac{1}{6} x-1 $$

Step-by-Step Solution

Answer

The solution is \(x < -18\) or \((-\infty, -18)\).

Step 1: Eliminate Fractions by Multiplying Both Sides

To eliminate the fractions, we find the Least Common Denominator (LCD) of 3 and 6, which is 6. Multiply every term of the inequality by 6:\[6 \left( \frac{1}{3}x + 2 \right) < 6 \left( \frac{1}{6}x - 1 \right)\]This simplifies to:\[2x + 12 < x - 6\]

Step 2: Isolate Variable Terms on One Side

To isolate the variable terms, subtract \(x\) from both sides:\[2x - x + 12 < x - x - 6\]This results in:\[x + 12 < -6\]

Step 3: Solve for the Variable

Now isolate \(x\) by subtracting 12 from both sides:\[x + 12 - 12 < -6 - 12\]This simplifies to:\[x < -18\]

Step 4: Express the Solution in Interval Notation

The solution to the inequality \(x < -18\) is expressed in interval notation as:\[(-\infty, -18)\]

Step 5: Graph the Solution Set

To graph \((-\infty, -18)\), draw a number line. Mark \(-18\) with an open circle to indicate that it is not included, and shade to the left to show all the numbers less than \(-18\).

Key Concepts

Interval Notation

Interval notation is a way of expressing a set of numbers along a number line. It is particularly helpful because it conveys which numbers are included and excluded in a concise manner.
When you see an inequality like \(x < -18\), the solution set includes all numbers less than \(-18\). In interval notation, this is written as \((-fty, -18)\).
Here, the negative infinity symbol \(-\infty\) indicates that the solution extends indefinitely to the left on the number line. Parentheses \(()\) are used rather than brackets \([]\) because \(-\infty\) and \(-18\) are not part of the solution. This notation tells us that the values are approaching but never actually reaching \(-\infty\) or \(-18\).

• The left part of the interval, \(-\infty\), implies going endlessly far into negative numbers.
• The right part of the interval, \(-18\), signals that the endpoint is not included in the solution set since there is an open circle at \(-18\) on the graph.

Least Common Denominator

When dealing with fractions in equations or inequalities, it's often useful to clear them out by finding a least common denominator (LCD). The LCD is the smallest number that both denominators can divide into evenly, and it helps to simplify expressions.
In the inequality \(\frac{1}{3}x + 2 < \frac{1}{6}x - 1\), the denominators are 3 and 6, making the LCD 6. By multiplying each term in the inequality by this common denominator, we remove the fractions completely:

• Multiply the term \(\frac{1}{3}x\) by 6 to get \(2x\).
• Multiply the term \(2\) by 6 to get \(12\).
• Multiply the term \(\frac{1}{6}x\) by 6 to get \(x\).
• Multiply the term \(-1\) by 6 to get \(-6\).

This yields the simplified inequality of \(2x + 12 < x - 6\). Eliminating fractions simplifies the problem and allows us to focus on the inequality itself without the complexities of fractional arithmetic.

Graphing Inequalities

Graphing inequalities is a visual way of representing solution sets on a number line. It shows which numbers make the inequality true.
For \(x < -18\), we start with a number line. Here’s how to graph it:

• Identify the critical number, which in this case, is \(-18\).
• Place an open circle on \(-18\) on the number line to show that \(-18\) itself is not included in the solution. An open circle means the endpoint is excluded.
• Shade the line to the left of \(-18\) to indicate that all numbers smaller than \(-18\) satisfy the inequality.

The shaded portion represents the infinite set of solutions that satisfy \(x < -18\). This approach gives a quick, intuitive view of the range of possible solutions, helping to solidify the understanding of inequalities.