Problem 18 - College Algebra with Corequisite Support

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: Understand the Inequality

The given inequality is \( |2x + 1| + 1 \leq 6 \). We want to isolate the absolute value expression first. To do this, subtract 1 from both sides of the inequality.

Step 2: Isolate the Absolute Value

Subtract 1 from both sides of the inequality to get \( |2x + 1| \leq 5 \). Now we have the absolute value isolated.

Step 3: Split the Absolute Value Inequality

The inequality \( |a| \leq b \) can be split into two linear inequalities: \( -b \leq a \leq b \). Apply this to our expression: \(-5 \leq 2x + 1 \leq 5 \).

Step 4: Solve the First Inequality

Solve \(-5 \leq 2x + 1\) by subtracting 1 from both sides to get \(-6 \leq 2x\). Then divide both sides by 2, resulting in \(-3 \leq x\).

Step 5: Solve the Second Inequality

Solve \(2x + 1 \leq 5\) by subtracting 1 from both sides to get \(2x \leq 4\). Then divide both sides by 2, resulting in \(x \leq 2\).

Step 6: Combine the Results

Combine the solutions from steps 4 and 5 to get \(-3 \leq x \leq 2\). This is the solution to the inequality.

Step 7: Write in Interval Notation

The solution \(-3 \leq x \leq 2\) in interval notation is \([-3, 2]\). This expresses all the values of \(x\) that satisfy the inequality.

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real numbers. It is often used to describe the solution set for inequalities. Instead of listing specific elements, you use intervals to show all numbers between two endpoints.

In interval notation:

• Brackets, such as [ or ], signify that an endpoint is included (closed interval).
• Parentheses, such as ( or ), indicate that an endpoint is not included (open interval).

The exercise solution \(-3 \leq x \leq 2\) is expressed in interval notation as \([-3, 2]\). This means the set of all numbers from \(-3\) to \(2\), including the endpoints \(-3\) and \(2\). Interval notation provides a concise and standardized format to express ranges.

Linear Inequalities

Linear inequalities, much like linear equations, involve algebraic expressions in which the variable is raised to the power of one, hence 'linear'. They state that one expression is either less than, greater than, less than or equal to, or greater than or equal to another.

For example, the inequalities \-6 \leq 2x\ and \2x \leq 4\ are linear. Solving them involves similar techniques to solving equations, such as adding, subtracting, multiplying, or dividing both sides by a number. However, remember when you multiply or divide by a negative number, you must flip the inequality sign.

In this exercise, we handled linear inequalities by isolating \(x\) and determining the permissible range of values that satisfy both inequalities. Each linear inequality provides a portion of the complete solution when combined.

Solving Inequalities

Solving inequalities is a fundamental skill in algebra. It involves finding all possible values of the variable that satisfy the conditions laid out by the inequality.

Let's recap the basic steps:

• Isolate the absolute value: Begin by isolating the absolute value expression by performing mathematical operations on both sides of the inequality.
• Split the inequality: For an inequality \(|a| \leq b\), write two inequalities: \(-b \leq a \leq b\).
• Solve each inequality separately: Treat each inequality as its linear equation counterpart, solving to find the range of \(x\), and don't forget the rule about flipping the inequality sign when multiplying or dividing by a negative number.
• Combine solutions: Integrate the results of the solved inequalities to get a comprehensive range of solutions.
• Express in interval notation: Finally, translate the solution into interval notation to clearly identify the range of valid \(x\) values.

Mastering these steps allows you to handle a variety of linear and absolute value inequalities efficiently.