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Chapter 1: Problem 75

Use interval notation to express the solution set of each inequality. $$|x-4| \leq 0$$

Short Answer

Expert verified
The solution set for the inequality \( |x-4| \leq 0 \) is [4,4]

Step by step solution

01

Understand the properties of absolute value

The absolute value of a number can be defined as its distance from zero on the number line. Therefore, the absolute value of any number is always non-negative, meaning it can either be zero or more than zero. However, it's never less than zero.
02

Solve the inequality

Let's observe the given inequality \(|x-4| \leq 0\). The absolute value of \(x-4\) is less than or equal to zero. Since we know absolute value can't be less than zero, the expression inside the absolute value must be zero. So, the inequality turns into \(x-4 = 0 => x = 4\)
03

Write solution in interval notation

The interval notation that represents the solution \(x = 4\) is [4,4]. This means x can only take the value 4 but no other values.

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Key Concepts

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

Absolute Value
Absolute value represents the distance of a number from zero on a number line, without considering the direction. It is always non-negative because distances cannot be negative. For example, the absolute value of both -7 and 7 is 7, represented as \( |-7|=7 \) and \( |7|=7 \). This property is fundamental when solving equations and inequalities involving absolute values, providing insight into the geometric meaning behind the number.
Inequalities
Inequality statements represent the relation between two values or expressions that may not be equal. They are used to show if one value is greater, less, or equal to another. There are several types of inequalities, including:
  • \( x < y \): x is less than y
  • \( x > y \): x is greater than y
  • \( x \leq y \): x is less than or equal to y
  • \( x \geq y \): x is greater than or equal to y
When dealing with absolute value inequalities like \(|x-4| \leq 0\), it's important to recognize that because the value can never be negative, the only solution is where the value inside the absolute value equals zero.
Solution Set
The solution set is a collection of all values that satisfy an equation or inequality. For example, in the inequality \(|x-4| \leq 0\), the only solution occurs when \(x-4=0\), giving us \(x=4\). To express this as a solution set in interval notation, we use [4,4]. This notation indicates that the solution is a single value, where x can only be 4. It's important to understand that each solution set corresponds directly to the conditions defined by the inequality or equation being solved.
Number Line
A number line is a helpful tool in mathematics for visualizing numbers and their relationships. It is a straight line where each point represents a real number. Positive numbers extend to the right, negative numbers to the left, and zero is typically centered.
When solving absolute value inequalities, using a number line can illustrate how far values are from zero or another reference point. For example, on a number line, the point 4 would be marked, and since the inequality \(|x-4| \leq 0\) holds only at x = 4, it directly corresponds to that one position on the line, weakening any confusion about possible validation for other points.

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Most popular questions from this chapter

The cost in dollars to hire a house painter is given by the function $$C(h, g)=15 h+14 g$$ where \(h\) is the number of hours it takes to paint the house and \(g\) is the number of gallons of paint required to paint the house. Find \(C(18,11)\)

Use interval notation to express the solution set of each inequality. $$|2 x-9|<7$$

Use the properties of inequalities to solve each inequality. Write answers using interval notation. $$2 x+3<11$$

Solve each quadratic inequality. Use interval notation to write each solution set. $$6 x^{2}-4 \leq 5 x$$

The average daily minimum-tomaximum temperatures for the city of Palm Springs during the month of September are \(68^{\circ} \mathrm{F}\) to \(104^{\circ} \mathrm{F}\). What is the corresponding temperature range measured on the Celsius temperature scale?
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