Problem 20 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find and graph the solution set of each inequality. $$ 9-|3 x+3| > 0 $$

Step-by-Step Solution

Answer

The solution set is \(-4 < x < 2\), graphed with open circles at -4 and 2.

Step 1: Analyze the Inequality

The inequality given is \( 9 - |3x + 3| > 0 \). We need to isolate the absolute value expression and understand its implication. This inequality implies \( |3x + 3| < 9 \).

Step 2: Break Down Absolute Value Inequality

The inequality \( |3x + 3| < 9 \) can be written in two parts: 1. \( 3x + 3 < 9 \)2. \( 3x + 3 > -9 \).

Step 3: Solve Each Part of the Inequality

First, solve \( 3x + 3 < 9 \): Subtract 3 from both sides - \( 3x < 6 \)Divide by 3 - \( x < 2 \).Next, solve \( 3x + 3 > -9 \):Subtract 3 from both sides - \( 3x > -12 \)Divide by 3 - \( x > -4 \).

Step 4: Combine the Solutions

The solutions \( x < 2 \) and \( x > -4 \) can be combined to express the solution set \( -4 < x < 2 \).

Step 5: Graph the Solution Set

To graph \( -4 < x < 2 \), draw a number line and place open circles at \( x = -4 \) and \( x = 2 \). Shade the region between these points to indicate that \( x \) includes all numbers between but not including \(-4\) and \(2\).

Key Concepts

Inequality Graphing

Understanding how to graph inequalities is a crucial skill in solving absolute value inequalities. Graphing helps visualize all possible solutions for an inequality.
A number line is a convenient tool for this task, particularly because it provides clear visual guidance.
- Start by identifying key points on the number line based on inequality solutions.- For "less than" or "greater than" constraints, use open circles. This represents that the number at the circle is not part of the solution.- Use shading to indicate solutions between specified points. If the solution is compounded like "and," shade the area between your critical points.- The example of the inequality \(-4 < x < 2\) involves open circles at both \(-4\) and \(2\), with shading in between.By following these steps, the graph gives a clear, visual representation of all numbers that satisfy the inequality.

Solving Inequalities

Solving inequalities involves finding the set of values that make an inequality true. When working with absolute value inequalities, it involves breaking the inequality into two possible scenarios. These scenarios stem from the definition of absolute value, which measures how far a number is from zero on the number line, without considering direction.
- For example, when solving \(|3x + 3| < 9\), recognize that this means \(-9 < 3x + 3 < 9\).- This gives rise to two separate inequalities that need solving: \(3x + 3 < 9\) and \(3x + 3 > -9\).In splitting the inequality:- You solve each part independently, treating them like standard linear equations.- Remember to perform the same operations on each side consistently—adding, subtracting, multiplying, or dividing.- In the example given, solving these resulted in \(x < 2\) and \(x > -4\).By systematically solving each part, you can find the complete range of solutions for the original inequality.

Combining Solutions

The process of combining solutions in inequality problems ensures that you appropriately express the range of values that satisfy all conditions stated in the inequality. This is crucial when dealing with absolute value inequalities that split into two separate solutions.
- Take the solutions from separately solved components and determine the final set of values that satisfy both. For absolute value inequalities, the solutions often intersect and yield a combined range.- In our example, we obtained \(x < 2\) from one part and \(x > -4\) from the other. Combining these gives us \(-4 < x < 2\), because the solutions overlap.When combining solutions:- Consider whether the solutions should intersect (as in "and" inequalities) or cover a wider span (as in "or" inequalities).- Clearly express this combined solution using inequality notation.This final step synthesizes the information, providing a clear and complete answer for which values of \(x\) meet all criteria of the inequality.