Problem 47 - Thomas Calculus

Question

If \(|x| \leq 3\) and \(x>-1 / 2,\) what can you say about \(x ?\)

Step-by-Step Solution

Answer

The solution is \(x \in (-\frac{1}{2}, 3]\).

Step 1: Understand the Absolute Value Inequality

The given inequality is \(|x| \leq 3\). We know that \(|x|\) represents the distance of \(x\) from zero on the number line. Thus, \(|x| \leq 3\) implies that \(x\) is in the interval \([-3, 3]\).

Step 2: Analyze the Second Inequality

The second part of the problem states \(x > -\frac{1}{2}\). This means \(x\) is greater than \(-\frac{1}{2}\).

Step 3: Find the Intersection of Intervals

Combine the two conditions: \([-3, 3]\) from the absolute value inequality and \(( -\frac{1}{2}, +\infty)\) from the second inequality. The solution to the combined inequalities is the intersection of these two intervals, which yields \((-\frac{1}{2}, 3]\).

Step 4: State the Conclusion

The values that satisfy both \(|x| \leq 3\) and \(x > -\frac{1}{2}\) are all \(x\) such that \(-\frac{1}{2} < x \leq 3\).

Key Concepts

Interval Notation

Interval notation is a system used to represent sets of numbers along the number line. It clearly indicates the range of values that are included in the set. In interval notation, brackets and parentheses are used to show whether endpoints are included or excluded from the interval.

• Square brackets \[\[ \]\] indicate that an endpoint is included in the interval. For example, \([-3, 3]\) means that the interval includes the numbers -3 and 3.
• Parentheses \( ( ) \) indicate that an endpoint is not included. For example, \((-rac{1}{2}, +\infty)\) means that the interval includes numbers greater than -1/2 but does not actually include -1/2.

Using interval notation can help in quickly identifying the solution set to inequalities, such as the solution to \(|x| \leq 3\), which is \([-3, 3]\). Therefore, a basic understanding of interval notation helps in solving inequalities effectively.

Intersection of Intervals

When solving inequalities involving absolute values, we often need to find the intersection of two intervals. The intersection of intervals denotes the values that satisfy all conditions simultaneously. Imagine you have two sets of numbers represented on the number line. The intersection is the portion where the two sets overlap.
For example, consider the intervals \([-3, 3]\) and \((-\frac{1}{2}, +\infty)\). To find the intersection:

• First, identify the common region shared by both intervals.
• Interval \([-3, 3]\) includes all numbers from -3 to 3.
• Interval \((-\frac{1}{2}, +\infty)\) includes numbers greater than -1/2.

The intersection of these intervals is \((-\frac{1}{2}, 3]\), which is the set of values that are shared by both intervals. These shared values satisfy both conditions in the original inequality problem.

Number Line

A number line is a visual representation of numbers along a straight line. It's an excellent tool for understanding inequalities and absolute values. By plotting numbers on a number line, you can visualize both the intervals and the overlap between them.
To use a number line for inequalities:

• Mark important numbers, such as and endpoints, with dots. Solid dots indicate that a number is included in the interval, while open dots mean it is not included.
• Draw lines or arcs to represent the intervals. For example, an interval \([-3, 3]\) would be a line with solid dots at -3 and 3.
• Find the intersection by noting where the lines overlap.

Using a number line can make it easier to see the solution of inequalities, such as in our exercise, where the solution for \(|x| \leq 3\) and \((x > -1/2)\) is seen clearly as \((-\frac{1}{2}, 3]\).