Problem 8 - Algebra for College Students

Question

Solve each inequality and graph the solutions. \(|x-2|<4\)

Step-by-Step Solution

Answer

The solution is \(-2 < x < 6\), graphed as an open interval between \(-2\) and \(6\).

Step 1: Understand the Inequality

The given inequality is \(|x-2| < 4\). This is an absolute inequality which will result in two separate inequalities. Absolute inequalities translate into compound inequalities when solved.

Step 2: Set Up the Compound Inequality

Since it is a less than absolute inequality, we set up the compound inequality by removing the absolute value and setting up a range: \(-4 < x-2 < 4\).

Step 3: Solve the Left Part of the Inequality

Start with the inequality \(-4 < x - 2\). To isolate \(x\), add 2 to both sides: \(-4 + 2 < x\) which simplifies to \(-2 < x\).

Step 4: Solve the Right Part of the Inequality

Next, solve the inequality \(x - 2 < 4\). Add 2 to both sides: \(x - 2 + 2 < 4 + 2\) which simplifies to \(x < 6\).

Step 5: Combine the Solutions

Combine the two inequalities from Steps 3 and 4: \(-2 < x < 6\). This is the solution set for the inequality \(|x-2| < 4\).

Step 6: Graph the Solution

On a number line, draw an open circle at \(-2\) and another at \(6\). Shade the region between \(-2\) and \(6\) to show that \(x\) is between these two numbers but not inclusive of them.

Key Concepts

Compound Inequalities

Compound inequalities involve two separate inequalities combined in a way to describe a wider range of solutions. In the context of absolute inequalities, such as \(|x - 2| < 4\), they represent the combined range where the inequality holds true. To solve this, we break it down by removing the absolute part to form two inequalities, yielding a compound form.

• The expression \( |x - 2| < 4 \) splits into \-4 < x - 2 < 4\.
• This means x must be greater than -4 but less than 4 when adjusted for the starting point, x - 2.

This demonstrates how absolute inequalities naturally lead us to compound inequalities, guiding us towards the true bounds of x that satisfy the original inequality. Compound inequalities are crucial in defining situations where two conditions must be satisfied at once.

Solving Inequalities

Solving inequalities, especially those involving absolute values, means determining the range of values that make the inequality true. By addressing each part separately, we ensure that our final solution honors the conditions set by the inequality.
To solve \(-4 < x - 2 < 4\), follow these steps:

• First, handle the left part: \-4 < x - 2\. Add 2 to both sides to find \-2 < x\.
• Next, solve the right part: \x - 2 < 4\. Similarly, add 2 to both sides to determine \x < 6\.
• Combine the two results for a final solution of \-2 < x < 6\.

Remember, solving an inequality is about finding all the values of the variable that satisfy the original statement. This approach systematically isolates the variable, making sure all potential solutions are considered.

Graphing Inequalities

Graphing inequalities helps visualize the range of solutions in a simple and intuitive way. Once an inequality is solved, like \(-2 < x < 6\), representing it graphically makes it easier to understand where the solutions lie on a number line.
Here’s how to graph \(-2 < x < 6\):

• Draw a number line with appropriate markings, focusing on the region from -2 to 6.
• Place an open circle at -2 and another at 6, indicating these values are not included in the solution set.
• Shade the area between -2 and 6. This visually expresses the solution \-2 < x < 6\, showing that x can be any number between these limits but not including -2 and 6 themselves.

Graphing concrete solutions from inequalities offers a clear pictorial representation of all possible solutions, making abstract solutions more tangible.