Problem 51 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. $$ |3-x| \leq 5 $$

Step-by-Step Solution

Answer

Interval: \[ -2, 8 \]; Graph: Number line shaded between -2 and 8.

Step 1: Understand the Absolute Inequality

The inequality \( |3 - x| \leq 5 \) means that the expression inside the absolute value (\(3 - x\)) lies within the range -5 to 5.

Step 2: Convert to Compound Inequality

Rewrite the absolute inequality \( |3 - x| \leq 5 \) as a compound inequality: \(-5 \leq 3 - x \leq 5\).

Step 3: Solve the Compound Inequality

Break the compound inequality into two parts and solve: \- \(-5 \leq 3 - x\):\Add \( x \) to both sides: \(-5 + x \leq 3\)\Add 5 to both sides: \( x \leq 8\)\- For \( 3 - x \leq 5\):\Subtract 3 from both sides: \(-x \leq 2\)\Multiply by -1 and reverse the inequality: \( x \geq -2\).

Step 4: Combine the Results

Combine the two parts of the compound inequality: \(-2 \leq x \leq 8\).

Step 5: Graph the Solution Set

Draw a number line and shade the region between -2 and 8, including the end points.

Step 6: Write the Solution in Interval Notation

Express the solution set in interval notation: \[ -2, 8 \].

Key Concepts

absolute value

Absolute value represents the distance of a number from zero on the number line, regardless of direction.
It's always non-negative. The absolute value of a number, like \(|x|\), is written as \(|x| = x\) if \(x \geq 0\) and \(|x| = -x \) if \(x < 0\).
For example, \(|3| = 3\) and \(|-3| = 3\).
In our problem, \(|3 - x|\) represents the distance from \(3 - x\) to zero. We need to find values of \(x\) such that \(|3 - x| \leq 5\).

compound inequality

A compound inequality is an equation with two or more inequalities joined together by 'and' or 'or'.
For example, \(-5 \leq 3 - x \leq 5\) is a compound inequality resulting from our absolute value problem. It means we are looking for \(x\) values satisfying both \(-5 \leq 3 - x\) and \(3 - x \leq 5\) simultaneously.
Breaking it down, we solve:

• \(-5 \leq 3 - x\) leading to \(-2 \leq x\)
• and \(3 - x \leq 5\) leading to \(x \leq 8\)

Combining we get: \(-2 \leq x \leq 8\).

interval notation

Interval notation is a simple way of writing the set of solutions to an inequality.
In interval notation, we use brackets and parentheses. Here are the rules:

• \r Square brackets \([ ]\) mean the boundary numbers are included.
• Round brackets \( ( )\) mean the boundary numbers are not included.

For \(-2 \leq x \leq 8\), both -2 and 8 are included in the solution set, so we use square brackets: \[ -2, 8 \].

graphing inequalities

Graphing inequalities involves shading the solutions on a number line. Start by drawing a number line and marking the critical points which are -2 and 8.
Since our interval \([-2, 8]\) includes both endpoints, use solid dots or closed circles on -2 and 8.
Then, shade the region between these points to show all the numbers \(-2 \leq x \leq 8\).
This graph visually represents our solution set, making it easier to understand and interpret.