Problem 62 - Algebra for College Students

Question

Solve each equation and inequality by inspection. \(|x+9|>-6\)

Step-by-Step Solution

Answer

The solution is all real numbers.

Step 1: Understand Absolute Values

An absolute value represents the distance of a number from zero on the number line, which is always non-negative. Therefore, \(|x+9|\) is always greater than or equal to zero.

Step 2: Analyze the Inequality

The inequality given is \(|x+9| > -6\). Since absolute values are always non-negative or zero, \(|x+9|\) is always greater than -6 for any real number as -6 is negative.

Step 3: Conclude the Solution

Since \(|x+9|\) is greater than any negative number, this inequality holds true for all real numbers. Therefore, there are no restrictions.

Key Concepts

Inequalities

Inequalities are expressions that show the relationship between two values, typically using symbols like \(<, >, \leq,\) and \(\geq\). In the context of absolute values, inequalities describe the range of values a variable can take. In the problem given, we analyze \(|x + 9| > -6\).
This inequality involves an absolute value, which represents a distance. It states that this distance is greater than \(-6\). Since the absolute value is always non-negative, \(|x + 9|\) is naturally greater than any negative number, thus making the inequality always true.
While understanding inequalities, pay attention to these points:

• Identify the expression and compare using the inequality sign.
• If you have absolute values, remember they can never be negative.
• Consider all possibilities of the variable values that satisfy the inequality.

By grasping these fundamentals, the concept of inequalities becomes more intuitive and manageable.

Real Numbers

The set of real numbers includes all the numbers that can be found on the number line, including both rational and irrational numbers.

• Rational numbers, like fractions and integers, can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers.
• Irrational numbers cannot be expressed as such a fraction, e.g., \(\pi\) and \(\sqrt{2}\).

When solving inequalities like \(|x + 9| > -6\), it's important to note that \(x\) can be any real number. This comes from understanding that the absolute value results will encompass all real numbers as potential solutions, as the inequality holds true universally.
Remember to:

• Consider all types of numbers—including decimals and fractions—when identifying the potential solutions.
• Leverage the property that real numbers are dense on the number line, meaning there are infinitely many real numbers between any two values.

Grasping real numbers helps unlock a deeper understanding of equations and inequalities.

Non-Negative Values

Non-negative values are numbers that are zero or positive.

• This includes all numbers that are either above zero or exactly zero.
• It's important to distinguish these from positive numbers, which strictly exclude zero.

The concept of absolute value is closely linked to non-negative values, as an absolute value is the distance of a number from zero and is non-negative by definition. This means that, regardless of whether a number is negative or positive, its absolute value will never be negative.
In the inequality \(|x + 9| > -6\), the term \(|x + 9|\) is always non-negative. Thus, it satisfies the inequality unconditionally because it will be always greater than \(-6\). Understanding the non-negative nature of absolute values simplifies analyzing and solving such inequalities.
When dealing with absolute values:

• Always remember they transform negative inputs to positive outputs and leave positive inputs unchanged.
• Know that this characteristic of zero or any positive value is key to solving inequalities involving absolute values.

By focusing on these points, you can effectively understand and apply the concept of non-negative values in mathematics.