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Chapter 1: Problem 114

Solve each equation or inequality. $$ |x-4|+5 \geq 4 $$

Short Answer

Expert verified
All real numbers.

Step by step solution

01

Isolate the Absolute Value Expression

Start by isolating the absolute value expression on one side of the inequality. Given \( |x-4| + 5 \geq 4 \), first subtract 5 from both sides: \[ |x-4| + 5 - 5 \geq 4 - 5 \] \[ |x-4| \geq -1 \]
02

Analyze the Absolute Value Inequality

Recall that the absolute value of any real number is always non-negative. Therefore, \( |x-4| \geq -1 \) is always true as absolute value is never less than a negative number.
03

State the Solution

Since \( |x-4| \geq -1 \) is always true for all real numbers, the solution includes all real numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Properties
Absolute value is a measure of distance from zero on a number line. It represents the magnitude of a number, regardless of its sign. For any real number
Inequality Solutions
Inequality solutions involve finding the set of values that satisfy a given inequality. These solutions can include finite or infinite ranges of values, or even all real numbers. The key steps are:
Analyze the inequality, solve for the variable, and ensure that the result makes sense in the context of the problem. For absolute value inequalities often break the solution into different cases based on the definition of absolute value
Real Numbers
Real numbers include all the numbers on the number line, encompassing both rational and irrational numbers, positive and negative values, and zero. Real numbers are crucial in solving inequalities like the one in our exercise, as they define the complete set of possible solutions. When the solution includes 'all real numbers,' it means every point on the number line is valid.

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