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Magazine Chapter 10: Problem 23
Solve each inequality. Write the solution set in interval notation and graph it. $$ \frac{1}{x}<2 $$
Short Answer
Expert verified
The solution set is \( ( -\infty, 0 ) \cup ( \frac{1}{2}, \infty ) \).
Step by step solution
01
Understanding the Inequality
We need to solve the inequality \( \frac{1}{x} < 2 \). This means we are looking for values of \( x \) such that the reciprocal of \( x \) is less than 2.
02
Solving the Inequality
Rewrite the inequality \( \frac{1}{x} < 2 \) as \( \frac{1}{x} - 2 < 0 \). Finding a common denominator, we have \( \frac{1 - 2x}{x} < 0 \). This inequality holds when the numerator and the denominator have opposite signs.
03
Numerator Negative, Denominator Positive
The expression \( \frac{1-2x}{x} < 0 \) indicates that either \( 1-2x < 0 \) and \( x > 0 \) or \( 1-2x > 0 \) and \( x < 0 \). We analyze these cases.
04
Analyzing Case 1
Solve \( 1-2x < 0 \). Subtract 1 from both sides: \( -2x < -1 \). Divide by \(-2\): \( x > \frac{1}{2} \). Since this is combined with \( x > 0 \), the solution from this case is \( x > \frac{1}{2} \).
05
Analyzing Case 2
Solve \( 1-2x > 0 \). This means \( 2x < 1 \); dividing by 2 gives \( x < \frac{1}{2} \). However, \( x < 0 \) must also hold true, giving \( -\infty < x < 0 \). Since 0 is not a valid solution (as it would make the denominator 0), the solution is \( x < 0 \).
06
Combining Solutions
We combine the solutions from both cases: \( x > \frac{1}{2} \) and \( x < 0 \). Thus, the solution set in interval notation is \(( -\infty, 0 ) \cup ( \frac{1}{2}, \infty )\).
07
Graphing the Solution
On a number line, we shade the intervals \( x < 0 \) and \( x > \frac{1}{2} \). Use open circles at 0 and \( \frac{1}{2} \) to indicate they are not included in the solution set.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation provides a concise way to represent a range of numbers, which is particularly useful in inequalities. Instead of listing several numbers or describing them verbally, we can express an entire set of solutions with a single expression. In the example given, the solution to the inequality can be expressed as \[(-\infty, 0) \cup (\frac{1}{2}, \infty)\] This tells us that the possible values of \(x\) are those that are less than 0 or greater than \(\frac{1}{2}\).
- Parentheses \(( )\) indicate that the endpoint is not included in the solution set.
- Infinity \((\infty)\) is always accompanied by parentheses, as it represents a bound that cannot be reached.
- The union symbol \((\cup)\) connects different parts of the interval, meaning both parts are solutions.
Reciprocals
Reciprocals play a vital role in inequalities, especially when fractions are involved. A reciprocal of a number is 1 divided by that number. For any nonzero number \(x\), its reciprocal is \(\frac{1}{x}\).
In our example, the inequality \(\frac{1}{x} < 2\) required us to think about the relationships between numbers and their reciprocals:
In our example, the inequality \(\frac{1}{x} < 2\) required us to think about the relationships between numbers and their reciprocals:
- When \(x > 0\), \(\frac{1}{x}\) is a positive number. The smaller \(x\) is, the larger its reciprocal becomes, but it must remain less than 2.
- When \(x < 0\), \(\frac{1}{x}\) is a negative number. This affects the direction of the inequality.
- Zero cannot be a part of our solution as \(\frac{1}{0}\) is undefined.
Graphing Inequalities
Graphing provides a visual representation of the solution set of an inequality, making it easier to understand and verify solutions.
For the inequality in our example, the graph helps illustrate the solution set \[(-\infty, 0) \cup (\frac{1}{2}, \infty)\]To graph this:
For the inequality in our example, the graph helps illustrate the solution set \[(-\infty, 0) \cup (\frac{1}{2}, \infty)\]To graph this:
- Draw a number line.
- Place open circles at critical points where the inequality changes, such as at 0 and \(\frac{1}{2}\).
- Shade the regions of the number line corresponding to \( x < 0 \) and \( x > \frac{1}{2} \).
