Problem 55 - Precalculus

Question

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-3| \leq 4$$

Step-by-Step Solution

Answer

The interval is \([-1, 7]\).

Step 1: Understanding the Absolute Value Inequality

Given the inequality \(|x-3| \leq 4\), it means the distance between \(x\) and 3 on the number line is less than or equal to 4. This can be interpreted as the numbers \(x\) that are within 4 units from 3.

Step 2: Break Down the Inequality

The inequality \(|x-3| \leq 4\) can be rewritten as two inequalities: \ \(x-3 \leq 4\) and \(x-3 \geq -4\). These are the boundary conditions for \(x\).

Step 3: Solve Each Inequality Separately

Solve the inequality \(x-3 \leq 4\) by adding 3 to both sides which gives \(x \leq 7\). \ Solve the inequality \(x-3 \geq -4\) by adding 3 to both sides which gives \(x \geq -1\).

Step 4: Identify the Solution Interval

Combining both solutions, the set of \(x\) that satisfies both \(x \leq 7\) and \(x \geq -1\) is \(-1 \leq x \leq 7\). This is the interval \([-1, 7]\).

Step 5: Represent the Solution on a Number Line

On a number line, shade the region from -1 to 7, including both endpoints since the inequality is \(\leq\). Use a solid dot at both \(-1\) and \(7\) to indicate that these values are included in the solution set.

Key Concepts

Inequality Solution

The requirement here is to solve an absolute value inequality. The specific inequality is \(|x-3| \leq 4\), which involves finding all real numbers \(x\) whose distance from 3 is at most 4 units. This interpretation is crucial as it sets the boundary conditions for your solution.

We start by breaking down the absolute value inequality into two separate inequalities, because an absolute value inequality like this one can be split. A key thing to remember is:

• \(|A| \leq B\) can be written as: \(-B \leq A \leq B\).

Using this property, you can express the initial inequality \(|x-3| \leq 4\) as:

• \(x - 3 \leq 4\)
• \(x - 3 \geq -4\)

Solving these linear inequalities separately allows us to find the values of \(x\). When you solve:

\(x - 3 \leq 4\) using addition, you add 3 to both sides to get \(x \leq 7\).

\(x - 3 \geq -4\), similarly adding 3, gives us \(x \geq -1\).

This leads us to conclude that the solution for \(x\) lies between -1 and 7, inclusive. Thus, the inequality solution set is derived as \([-1, 7]\).

Number Line Representation

Once we have the solution for the inequality, representing it on a number line provides a visual cue that reinforces understanding. Let's consider our solution set: \([-1, 7]\). This means that our solution includes all numbers from -1 to 7, and both -1 and 7 themselves.

Here's how to represent this on a number line:

• Draw a horizontal line, which represents the number line.
• Mark the points -1 and 7 with solid dots to show that these endpoints are included.
• Shade the entire region between -1 and 7. This shaded region depicts all values \(x\) that satisfy the inequality \(-1 \leq x \leq 7\).

Solid dots (also known as closed circles) at -1 and 7 specifically indicate that these points are part of the solution set. Hence, this graphical representation accurately visualizes the solution on the number line.

Interval Notation

Interval notation provides a succinct way of writing the set of solutions for inequalities. After resolving an absolute value inequality like \(|x-3| \leq 4\), the solutions can be effectively communicated through this notation.

The interval \([-1, 7]\) describes all numbers from -1 to 7, including the endpoints. The brackets "[" and "]" show that the values -1 and 7 are part of the solution. If the endpoints were not included, we would use parentheses "(" and ")" instead. Here's an easy guide:

• Square brackets \([]\) imply that the endpoint is included and the inequality is "less than or equal to" or "greater than or equal to".
• Round brackets \(( )\) indicate the endpoint is not included, suitable for strict inequalities like "less than" or "greater than" only.

In this exercise, since \(-1 \leq x \leq 7\) includes both endpoints, the interval notation is \([-1, 7]\). This method is compact and efficient for conveying the set of numbers that satisfy the given inequality.