Problem 55 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|10-4 x| \geq-4$$

Step-by-Step Solution

Answer

All real numbers.

Step 1: Understand the Inequality

The given inequality is \(|10-4x| \geq -4\). Recall that the absolute value of any expression is always non-negative. Therefore, the absolute value will always be greater than or equal to any negative number.

Step 2: Solve the Absolute Value Inequality

Since the absolute value, \(|10-4x|\), is always non-negative and \(-4\) is negative, \(|10-4x| \geq -4\) is always true for all real numbers.

Step 3: State the Solution

Thus, the inequality is satisfied for all real numbers \(x\).

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics. It measures the distance of a number from zero on the number line, regardless of direction. For any real number \(a\), the absolute value is denoted by \(|a|\). This means:

• |a| is always a non-negative value.
• |a| equals a if a is non-negative (i.e., a follows: \(a \ge 0\).
• |a| equals \(-a\) if a is negative (i.e., a follows: \(a < 0\).

For example, \(|3| = 3\) and \(|-3| = 3\). Absolute value treats every number as its positive counterpart.

Inequalities

Inequalities are statements that compare two values or expressions, showing if one is greater, less, or equal under certain conditions. They play a crucial role in solving problems with ranges of possible solutions. Common inequality symbols are:

• \(\textless\textbackslashtextgreater\) - less than/greater than
• \(\textleq\textbackslashtextgeq\) - less than or equal to/greater than or equal to

Remember, when multiplying or dividing inequalities by a negative number, the inequality sign flips direction. Inequalities help in understanding ranges of solutions rather than single values.

Real Numbers

Real numbers include all rational and irrational numbers, covering any number along the infinite number line. This includes:

• Whole numbers like 0, 1, 2
• Fractions like 1/2
• Decimals like 3.14
• Irrational numbers like \(\pi\)

The set of real numbers is represented by \(\textbackslashmathbb\textbackslashR\). They form the foundation of most calculations encountered in algebra and everyday math.