Problem 72 - Beginning and Intermediate Algebra

Question

The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$0 \leq|4 a+1|$$

Step-by-Step Solution

Answer

The solution to the inequality \(0 \leq |4a + 1|\) is \((-∞, +∞)\) which represents all real numbers along the number line.

Step 1: Analyze the absolute value inequality

We are given the inequality \(0 \leq |4a + 1|\). Absolute value represents the distance from a number to zero, meaning it will always be non-negative. The inequality states that the distance of the value of \(4a + 1\) from zero is greater than or equal to zero, which is true for any value of "a".

Step 2: Remove the absolute value

Since the inequality is true for any value of "a", we do not need to split the inequality into cases because there are no restrictions to the value of "a".

Step 3: Write the interval notation

Since all values of "a" satisfy the inequality, we can write the solution set using interval notation: \((-∞, +∞)\) This represents all real numbers along the number line. #Answer#: \((-∞, +∞)\)

Key Concepts

Set Notation

Set notation is a way to describe a collection of elements, often numbers, by specifying properties that the elements must satisfy. It offers a concise and precise way to express sets, avoiding lengthy descriptions. In mathematics, especially in solving equations, set notation presents solutions in a manner that clearly communicates the conditions each element has to meet.

When writing a solution in set notation, the format often uses curly braces \"{}\". Everything inside these braces is part of the set. For example, the equation solution might be expressed as \{ x \, | \, x = 2 \}\, meaning the set includes only the number 2.

• In the context of this exercise, set notation would describe solution sets for equations where specific conditions need specifying.
• For instance, \{ a \, | \, a \, \text{meets the condition} \}\ tells us which values of \(a\) satisfy the given scenario.

Set notation is particularly useful for clearly conveying solution sets in an unambiguous way.

Interval Notation

Interval notation is a shorthand used in mathematics to describe a range of values on the number line. It stands out for its efficiency in expressing continuous sets of numbers, mainly used for inequalities.

Intervals can be described as bounded (having both start and end points) or unbounded (extending infinitely in one or both directions). For example:

• \([a, b]\) describes the set of all numbers \(x\) such that \(a \leq x \leq b\).
• \((a, b)\) includes all numbers greater than \(a\) and less than \(b\), excluding \(a\) and \(b\) themselves.
• \([a, \infty)\) means the set of all numbers starting from \(a\) and extending indefinitely.

In the original exercise, the solution of \((-\infty, +\infty)\) showcases that any real number satisfies the condition due to the nature of absolute values. Thus, interval notation efficiently represents this broad range of solutions.

Linear Inequalities

Linear inequalities are expressions involving linear functions that use inequality symbols such as \(<\), \(>\), \(\leq\), and \(\geq\). Solving linear inequalities involves finding the set of values that make the inequality true.

To solve linear inequalities, we often use similar techniques to solving linear equations but with careful attention to the direction of the inequality when performing operations like multiplication or division by a negative number. This may require flipping the inequality sign.

• For example, solving \(3x + 2 < 11\) involves subtracting 2 from both sides and then dividing by 3.
• The solution might be expressed as \(x < 3\).

These inequalities can have solutions that form ranges, and using interval notation provides clarity. Understanding linear inequalities is fundamental in managing more complex problems like those involving absolute values. In the exercise provided, understanding how \(|4a + 1|\) fits into the inequalities assists in distinguishing when such inequalities hold true for all real numbers, as it does in the interval \((-\infty, +\infty)\).