Problem 20 - College Algebra

Question

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation. $$ |-2 x+7| \leq 13 $$

Step-by-Step Solution

Answer

\([-3, 10]\)

Step 1: Understanding the Inequality

We start with the inequality \(|-2x+7| \leq 13\). An absolute value inequality \(|A| \leq B\) implies that \(-B \leq A \leq B\). In this case, \(A\) is \(-2x+7\) and \(B\) is 13.

Step 2: Setup the Compound Inequality

Based on the previous step, rewrite the inequality without the absolute value as a compound inequality: \[-13 \leq -2x + 7 \leq 13\]

Step 3: Solve the First Inequality

Start with the inequality \(-13 \leq -2x + 7\). Subtract 7 from both sides to isolate the term with \(x\): \[-13 - 7 \leq -2x\] \[-20 \leq -2x\] Now divide both sides by -2, remembering to reverse the inequality sign: \[x \leq 10\]

Step 4: Solve the Second Inequality

Now solve \(-2x + 7 \leq 13\). Subtract 7 from both sides: \[-2x \leq 6\]Then, divide by -2 and reverse the inequality sign: \[x \geq -3\]

Step 5: Combine the Solutions

We now combine the solutions from the two inequalities: \[-3 \leq x \leq 10\]This can be written in interval notation as \([-3, 10]\).

Key Concepts

Compound Inequality

A compound inequality is a statement that involves two inequalities combined by the word "and" or "or." In the case of absolute value inequalities, we often end up with a compound inequality. This happens because the absolute value of an expression is essentially the distance from zero, and it can be positive or negative.

For example, considering the inequality \(|-2x + 7| \leq 13\), we can split this into two parts:

• The left part: \(-13 \leq -2x + 7\)
• The right part: \(-2x + 7 \leq 13\)

The "and" in this situation indicates that both conditions must be true at the same time. This results in a solution for the variable \(x\) that satisfies both inequalities.

Understanding how to break down and solve a compound inequality is key, as it provides us with a range where the solutions lie.

Interval Notation

Interval notation is a mathematical method of writing down a set of numbers that includes all the values within a certain range. It includes two numbers, which are called endpoints, and defines when these endpoints are part of the solution.

There are two types of brackets used in interval notation:

• Square brackets [ ]: These indicate that the endpoint is included in the interval (inclusive).
• Parentheses ( ): These indicate that the endpoint is not included (exclusive).

For instance, once we solve the compound inequalities, we find that \(x\) lies between \(-3\) and \(10\). Using interval notation, this is expressed as \([-3, 10]\).- The square brackets signify that both -3 and 10 are included as solutions.

This concise form makes it easy to grasp the range of possible values \(x\) can take.

Solving Inequalities

Solving inequalities involves finding all possible values of the variable that satisfy the inequality. With absolute value inequalities, the process consists of:

• Understanding and rewriting the inequality without absolute values as a compound inequality.
• Solving each part of the compound inequality separately.
• Reversing the inequality sign when multiplying or dividing by a negative number.
• Combining solutions to express a singular range of possible values.

For instance, we start with \([-13 \leq -2x + 7 \leq 13]\) and split it:- Solving \(-13 \leq -2x + 7\) gives \(x \leq 10\).
- Solving \(-2x + 7 \leq 13\) results in \(x \geq -3\).

Solving inequalities properly ensures that we account for all potential values that \(x\) can take within the given range defined by the original equation. Always remember to handle signs carefully, especially with absolute values, to avoid errors.