Problem 18 - College Algebra

Question

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation. $$ |2 x+1|+1 \leq 6 $$

Step-by-Step Solution

Answer

The solution in interval notation is \([-3, 2]\).

Step 1: Isolate the Absolute Value

The first step is to isolate the absolute value expression. Start by moving 1 to the other side of the inequality: \[ |2x + 1| + 1 \leq 6 \] Subtract 1 from both sides:\[ |2x + 1| \leq 5 \]

Step 2: Set Up Compound Inequality

When dealing with an inequality involving an absolute value, we create a compound inequality representing both possible scenarios for the absolute value. Therefore, we consider: \[ -5 \leq 2x + 1 \leq 5 \]

Step 3: Solve Left Side of Inequality

Solve the left side of the inequality: \[ -5 \leq 2x + 1 \] Subtract 1 from both sides: \[ -6 \leq 2x \] Divide both sides by 2: \[ -3 \leq x \]

Step 4: Solve Right Side of Inequality

Now solve the right side of the inequality: \[ 2x + 1 \leq 5 \] Subtract 1 from both sides:\[ 2x \leq 4 \] Divide both sides by 2:\[ x \leq 2 \]

Step 5: Combine Solutions

Combine the solutions obtained from the left and right sides of the compound inequality: \[ -3 \leq x \leq 2 \] This represents the interval for the solution.

Key Concepts

Interval Notation

Interval notation is a method of writing the set of solutions to an inequality in a concise and clear way. In interval notation, we describe a range of values for which an inequality holds true, using brackets and parentheses.

• Square brackets \([\ ]\) denote inclusive boundaries, meaning the endpoint is part of the solution set.
• Parentheses \((\ )\) indicate that the endpoint is not included in the solution set.

For example, the solution obtained in the step-by-step process, \(-3 \leq x \leq 2\), means that \(x\) can take any value from \(-3\) to \(2\), inclusive. In interval notation, this is written as:\[[-3, 2]\]This means our solution set is the continuous range of numbers starting from \(-3\) to \(2\), where both endpoints are included. This format is widely used because it is clean, efficient, and visually shows the range of solutions.

Compound Inequality

A compound inequality combines two inequalities into one statement using the word 'and' or 'or'. In the context of absolute value inequalities, we examine both the positive and negative scenarios of the expression inside the absolute value.

For the inequality \(|2x + 1| \leq 5\), we set up a compound inequality because the absolute value expression can be equal to either the positive 5 or negative 5 values:
\[-5 \leq 2x + 1 \leq 5 \]This compound inequality requires that \(2x + 1\) falls between \(-5\) and \(5\).

Solving this type of inequality involves transforming it into two separate inequalities and then solving each one individually. What's unique about a compound inequality like this is that the solutions from both parts will be combined into one continuous set, helping to maintain both conditions simultaneously.

Solving Inequalities

Solving inequalities involves finding the set of all possible values that satisfy the given inequality. For absolute value inequalities, this process includes handling two potential scenarios: one for the expression inside the absolute value being positive and one for it being negative.

Here’s the step-by-step breakdown for solving an inequality:

• Isolate the absolute value expression: Begin by moving all other terms to the other side of the inequality.
• Set up a compound inequality: Reflect the two scenarios—the expression inside the absolute value could be either negative or positive.
• Solve each part: Solve the separate inequalities like regular equations, maintaining the inequality direction unless you multiply or divide by a negative number, which flips the inequality.
• Combine the results: Compile the solutions to form the solution in interval format.

Each step involves maintaining the integrity of the inequality symbols, unlike equations that are generally manipulated equally on both sides, so handle each step carefully.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers along with operational symbols like plus, minus, multiply, and divide. They form the foundation of solving inequalities and equations.

An expression like \(2x + 1\) is an algebraic expression that shows multiplication of a variable \(x\) by 2, followed by an addition of 1.These expressions become central when solving inequalities because:

• They have to be manipulated carefully during steps to isolate variables.
• Understanding how to adjust these expressions correctly determines whether the inequality remains true.
• Each part impacts how the entire inequality is represented when rewritten.

Understanding how to work with algebraic expressions helps in visualizing the changes that occur throughout the process. Each operation should be executed with the ultimate goal of isolating the variable efficiently. Thus, mastering algebraic expressions is key to solving complex inequality problems.