Problem 36 - College Algebra

Question

For the following exercises, graph the function. Observe the points of intersection and shade the \(x\) -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. $$ |x-2|<7 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-5, 9)\).

Step 1: Understand the inequality

The inequality \(|x-2| < 7\) involves an absolute value. The absolute value is less than 7, indicating that the expression inside the absolute value is between -7 and 7.

Step 2: Set up the compound inequality

The inequality \(|x-2| < 7\) can be rewritten without the absolute value as two separate inequalities:1. \(x-2 < 7\)2. \(x-2 > -7\).

Step 3: Solve each inequality

First, solve \(x-2 < 7\):\[x < 7 + 2\]\[x < 9\]Then, solve \(x-2 > -7\):\[x > -7 + 2\]\[x > -5\].

Step 4: Combine the solutions

Combine the solutions from both inequalities to get the compound solution:\(-5 < x < 9\).

Step 5: Graph the solution

Graph the inequality on a number line: shade the region between \(x = -5\) and \(x = 9\), not including the endpoints since it is a strict inequality (<, >). Use open circles at -5 and 9.

Step 6: Write the solution in interval notation

Use interval notation to express the solution set:\((-5, 9)\). The parentheses indicate that the endpoints are not included in the solution.

Key Concepts

Compound Inequalities

When dealing with absolute value inequalities, such as \(|x-2| < 7\), we're often required to break them down into compound inequalities. This turns our problem into two simpler inequalities that must be solved together. In the case of absolute values, the expression inside is bounded by the positive and negative value of the given number.
For the inequality \(|x-2| < 7\), this means:

• \(x - 2 < 7\)
• \(x - 2 > -7\)

This step is crucial as it translates the absolute value inequality into two straightforward linear inequalities that describe the same range. Solving each of these provides the range of values that satisfy the original absolute value inequality.
Always keep in mind that solving compound inequalities involves finding common solutions to both inequalities.

Interval Notation

Once the compound inequalities are solved, the solution is often expressed using interval notation. This is a concise way to describe a range of numbers on the number line. For instance, the compound solution we derived is \(-5 < x < 9\).
In interval notation, we write this solution as \((-5, 9)\).

• Parentheses \( ( , ) \) are used to indicate that the endpoints are not included.
• If the endpoints were part of the solution, we would use brackets, \[ [ , ] \].

Be sure to use interval notation accurately to convey whether endpoints are part of the solution or not. This helps in understanding the precise range of numbers that the variable can assume.

Inequality Solutions

Solving inequalities requires clear steps to ensure accuracy. This involves isolating the variable, performing arithmetic operations, and ensuring the inequality's direction isn't altered unless necessary.
In the example \(|x-2| < 7\):

• Resolve \(x - 2 < 7\) to find \(x < 9\).
• Resolve \(x - 2 > -7\) to find \(x > -5\).

By combining these findings, \(-5 < x < 9\), you have the set of all possible values that satisfy the inequality. Attention to detail in performing operations like adding or subtracting the same number from both sides is essential to achieve the correct solution.

Graphing Inequalities

Graphing inequalities is an essential skill for visualizing solutions. It helps to clearly see which values satisfy the inequality. The solution \(-5 < x < 9\) can be graphed on a number line as follows:

• Draw a number line and mark points for -5 and 9.
• Place open circles on -5 and 9 to represent that these points are not included in the solution.
• Shade the line segment between -5 and 9 to indicate all the numbers within this range are solutions.

Remember, for strict inequalities (those with \(<\) or \(>\)), we use open circles. For inequalities with \(\leq\) or \(\geq\), filled circles indicate inclusion of the endpoint. This visual representation not only helps verify your solution but also assists in understanding the scope of variables in your equations.