Problem 3 - Precalculus Mathematics for Calculus

Question

(a) The solution of the inequality \(|x| \leq 3\) is the interval ________. (b) The solution of the inequality \(|x| \geq 3\) is a union of two intervals ___________ \(\bigcup\) _________.

Step-by-Step Solution

Answer

(a) [-3, 3]; (b) (-∞, -3] ∪ [3, ∞).

Step 1: Understanding Absolute Value Inequality

Absolute value inequalities describe the distance of a number from zero on the number line. The inequality \(|x| \leq 3\) means that the distance of \(x\) from zero should be less than or equal to 3.

Step 2: Solving the Inequality \(|x| \leq 3\)

The inequality \(|x| \leq 3\) can be split into two inequalities: \(x \leq 3\) and \(-x \leq 3\). Solving \(-x \leq 3\) gives \(x \geq -3\). These two inequalities combine to give the solution \(-3 \leq x \leq 3\). In interval notation, this is \([-3, 3]\).

Step 3: Understanding Absolute Value Greater Than Inequality

The inequality \(|x| \geq 3\) means the distance of \(x\) from zero should be greater than or equal to 3. This means \(x\) could be either 3 or greater, or it could be -3 or less.

Step 4: Solving the Inequality \(|x| \geq 3\)

The inequality \(|x| \geq 3\) splits into two: \(x \geq 3\) and \(-x \geq 3\). The inequality \(-x \geq 3\) simplifies to \(x \leq -3\). Thus, the solution is the union of two intervals: \((-\infty, -3]\) and \([3, \infty)\).

Key Concepts

Interval Notation

When we solve inequalities, we often use interval notation as a simple way to represent the set of solutions. Interval notation expresses a range of values, showing the start and end points of the interval. These points are wrapped in brackets or parentheses.

• Closed brackets [ ] mean the number is included in the interval. So, in the expression \([-3, 3]\), both -3 and 3 are part of the solution set.
• Open brackets ( ) mean the number is not included. For example, \( \(-\infty, -3\) \) means -3 is not included, and neither is \(-\infty\) since infinity is a concept rather than a concrete number.

Blending both styles allows us to express complex solutions neatly, such as unions of intervals. A union, depicted as \( \bigcup \), combines separate intervals into one solution, like \( (-\infty, -3] \bigcup [3, \infty) \). It shows the value of \(x\) could lie in either interval.

Inequality Solutions

Inequalities show the possible values that a variable can take. They can express boundaries for these values either in a range (for \(|x| \leq\) situations) or as separate intervals (for \(|x| \geq\) situations). A key step in solving inequalities is to split the absolute value inequality into two simpler inequalities.
For \(|x| \leq 3\), we interpret this as both \(x\) being less than or equal to 3 and greater than or equal to -3. This tells us that \(x\) falls within the bounds of -3 and 3, expressed in interval notation as \([-3, 3]\).
With \(|x| \geq 3\), the problem shows that \(x\) can either be greater than or equal to 3 or less than or equal to -3. So, we form two separate intervals and show them as a union: \( (-\infty, -3] \bigcup [3, \infty) \). This union reflects that \(x\) can be in one interval or the other.

Number Line Distance

Absolute value inequalities are crucial for understanding number line distance.
They measure how far a number is from zero on a number line, regardless of direction—positive or negative.
With inequalities like \(|x| \leq 3\), the number \(x\) is no farther than 3 units from zero. On a number line, you can visualize this with points between -3 and +3. Every point here satisfies the condition.
In contrast, \(|x| \geq 3\) considers all points outside this range, beyond -3 on the left and beyond 3 on the right. Thus, only points outside, extending infinitely in both directions, satisfy the inequality. This interval begins from the far left, stops at -3, and resumes from 3 extending infinitely to the right, precisely as described by \( (-\infty, -3] \bigcup [3, \infty) \). When graphed, the number line vividly shows the distances involved.