Problem 7 - Algebra 2

Question

Solve each inequality. Graph the solution set on a number line. $$ |h| < 3 $$

Step-by-Step Solution

Answer

The solution set is \(-3 < h < 3\). Graph it with open circles at -3 and 3, shading in between.

Step 1: Understanding Absolute Value Inequality

The inequality \(|h| < 3\) represents all values of \(h\) whose distance from 0 is less than 3 on a number line. That means \(h\) must be greater than -3 and less than 3.

Step 2: Breaking Down the Inequality

From \(|h| < 3\), we derive the two inequalities: 1. \(h < 3\) 2. \(h > -3\)Thus, the compound inequality becomes \(-3 < h < 3\).

Step 3: Solution Interval

The solution to the inequality \(-3 < h < 3\) includes all real numbers \(h\) that lie between -3 and 3, not including -3 and 3 themselves.

Step 4: Graphing the Solution

To graph this on a number line, place open circles at -3 and 3 to indicate these numbers are not included. Shade the region between -3 and 3 to represent all numbers in this interval. This graph visually represents the solution set \(-3 < h < 3\).

Key Concepts

Inequalities

Inequalities are mathematical expressions used to compare values. They show that one value is greater than, less than, equal to, or not equal to another value. Common inequality symbols include:

• \( > \) : Greater than
• \( < \) : Less than
• \( \geq \) : Greater than or equal to
• \( \leq \) : Less than or equal to

Absolute value inequalities like \(|h| < 3\) tell us about the distance from zero instead of distinct directional comparisons.
These inequalities help determine all possible values, revealing a range of solutions that satisfy the inequality. To solve, we turn them into compound inequalities that make it easier to interpret and solve.

Solution Set

The solution set is the collection of all values that satisfy a given inequality or equation. For the inequality \(|h| < 3\), the solution set consists of all numbers that have an absolute value less than 3.
This means any number between -3 and 3 meets this condition. The expression becomes a compound inequality: \(-3 < h < 3\). This notation tells us that the value of \(h\) must be greater than -3 and less than 3.
The solution set is continuous and includes all decimals, fractions, and real numbers that fit this range, except for the end values of -3 and 3 themselves. It's crucial to understand what numbers belong in the solution set to correctly apply or graph an inequality.

Graphing on a Number Line

Graphing an inequality's solution set on a number line provides a visual representation of the range of values that satisfy the inequality. To graph \(-3 < h < 3\), we:

• Place open circles on -3 and 3
• Shaded the segment between -3 and 3

The open circles indicate that these points are not included in the solution set because the inequality is strict (using \(<\) instead of \(\leq\)).
Shading between the points shows all real numbers that solve the inequality are part of the solution set. This visual makes it easier to understand which numbers are meant when solving or interpreting inequalities.

Compound Inequalities

Compound inequalities are two or more inequalities joined by the words "and" or "or." These compound forms help simplify and clarify expressions like absolute value inequalities. In the given problem, the compound inequality \(-3 < h < 3\) comes from the original \(|h| < 3\).
This form tells us that both conditions \(h > -3\) and \(h < 3\) must be satisfied simultaneously. Thus, the term "compound" refers to multiple conditions being true at once.
Understanding compound inequalities is essential because they often appear in both algebra and real-world problems, helping us see the limitations or ranges for certain variables.