Question

The inequality \(|x|>-0.5\) has all real numbers as solutions. Explain why.

Short answer

The inequality is true for all real numbers because the absolute value of any real number is always non-negative and thus greater than any negative number, including -0.5.

Step-by-step solution

Understand the Absolute Value

The absolute value \(|x|\) of a number x is the non-negative value of x without regard to its sign. Thus, \(|x|\) is always greater than or equal to 0.

Analyze the Given Inequality

The given inequality is \(|x| > -0.5\). Notice that -0.5 is a negative number.

Compare Absolute Value and Negative Number

Since \(|x|\) is always non-negative and any absolute value is greater than or equal to 0, \(|x|\) will always be greater than any negative number, including -0.5.

Conclude the Solution

Because \(|x|\) is always greater than -0.5 for any real number x, every real number satisfies the inequality \(|x| > -0.5\).

Key concepts

Absolute Value

The concept of absolute value is crucial in understanding many algebraic expressions. The absolute value of a number, denoted as \(|x|\), is the distance of that number from zero on the number line. It is always a non-negative value. For example, the absolute value of both 3 and -3 is 3. In general terms, \(|x| = x\) if \(( x \ge 0)\) and \(|x| = -x\) if \(( x < 0)\). This concept ensures that absolute value expressions are always positive or zero. It is important when dealing with inequalities because it simplifies the comparison of expressions.

Inequalities in Algebra

Inequalities play a significant role in algebra. They are used to represent a range of values that satisfy given conditions. Inequalities come in several forms, including <, >, ≤, and ≥. Solving inequalities involves finding the set of all possible values that make the inequality true.
Here’s a quick methodology for solving inequalities:

• Isolate the variable on one side of the inequality.
• Simplify the inequality as needed.
• Consider the effect of multiplying or dividing by a negative number, as this reverses the inequality sign.

For absolute value inequalities, the process involves breaking them into two separate inequalities that reflect the positive and negative scenarios of the absolute value expression.

Real Numbers

Real numbers are a fundamental set of numbers in mathematics. They include all rational and irrational numbers. Rational numbers are those that can be expressed as fractions, while irrational numbers cannot be expressed as simple fractions. Real numbers can be plotted on a number line, providing a visual representation of their order and magnitude.
The key takeaway is that real numbers encompass all possible values you might encounter in basic algebra and calculus. Hence, when dealing with inequalities like \(|x| > -0.5\), it’s essential to recognize that since the absolute value is always non-negative, every real number \(x\) would satisfy this inequality.