Problem 38 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ 7|x+2|+5>4 $$

Step-by-Step Solution

Answer

The solution is all real numbers, expressed as \((-\infty, \infty)\).

Step 1: Simplify the inequality

We start by isolating the absolute value expression. We have the inequality \(7|x+2| + 5 > 4\). First, subtract 5 from both sides to get: \[7|x+2| > -1\].Notice that any expression of the form \(|x+2|\) is always non-negative, so \(7|x+2|\) will always be greater than \(-1\). Thus, the inequality is always true for all real numbers \(x\).

Step 2: Conclusion

Since the inequality \(7|x+2| > -1\) holds for all real numbers \(x\), our solution is that any real number will satisfy the inequality.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. In mathematical terms, the absolute value of a number \( x \) is written as \( |x| \). It is always a non-negative value.
Absolute values are useful in inequalities to express ranges of values. For example:

• \(|x| \leq 3\) means \(-3 \leq x \leq 3\).
• \(|x| \geq 5\) indicates that \(x \geq 5\) or \(x \leq -5\).

In the exercise above, the inequality involves an absolute expression \(|x + 2|\). Since absolute values are always non-negative, \(|x + 2| \geq 0\), ensuring that \(7|x+2|\) is greater than or equal to zero as well. This fact simplifies inequalities involving absolute values.

Interval Notation

Interval notation is a way of representing a set of numbers, typically all the values between two endpoints. It's a concise way to depict continuous ranges of real numbers without listing all values.
Below are key symbols used in interval notation:

• Brackets \([ \text{and} ]\) represent inclusivity, meaning the endpoint is part of the interval.
• Parentheses \(( \text{and} )\) imply exclusivity, meaning the endpoint is not part of the interval.
• Example: \([-2, 3)\) denotes all numbers \(x\) such that \(-2 \leq x < 3\).

In the given problem, since the inequality holds for any real number, the solution in interval notation is \((-olinebreak[4]\infty, olinebreak[4]\infty)\). This states that every real number satisfies the inequality, making it an unbounded interval across the number line.

Real Numbers

Real numbers include all numbers that can be found on the number line. This encompasses rational numbers, such as integers and fractions, as well as irrational numbers that cannot be expressed as fractions.
Real numbers have critical properties in mathematics:

• They are used to measure continuous quantities.
• They include natural numbers, whole numbers, integers, and both rational and irrational numbers.
• Examples include \(0, -1, 2.5, \pi, \sqrt{2}\).

In the context of the exercise, when we determined that \(7|x+2| > -1\) is always true, it was referring to the property of real numbers where absolute value expressions are always non-negative. Therefore, any real number \(x\) will satisfy the inequality, showcasing the comprehensive nature of real numbers as a complete set for solutions in equations and inequalities.