Problem 18 - College Algebra

Question

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

Step 1: Isolate the Absolute Value

Start by isolating the absolute value expression. Subtract 1 from both sides of the inequality to get \(|2x + 1| \leq 5\).

Step 2: Split the Inequality

The expression \(|2x + 1| \leq 5\) can be split into two separate inequalities:1. \(2x + 1 \leq 5\)2. \(2x + 1 \geq -5\).

Step 3: Solve First Inequality

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

Step 4: Solve Second Inequality

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

Step 5: Combine the Solutions

Combine the solutions from Steps 3 and 4 into one solution set: the intersection of \(x \geq -3\) and \(x \leq 2\) is \(-3 \leq x \leq 2\).

Step 6: Write the Solution in Interval Notation

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

Key Concepts

Interval Notation

Interval notation is a method used to represent a set of numbers between two endpoints, often used in expressing the solution sets of inequalities. Instead of using words or long descriptions, interval notation presents a clear and concise way to show the solution set.
For example, if an inequality solution falls between a minimum and maximum value, like in our exercise where \(-3 \leq x \leq 2\), this can be simply communicated with interval notation as \([-3, 2]\).
There are rules for writing the endpoints in interval notation:

• Use square brackets, \([ \text{ and } ]\), to include the endpoint, signifying that the number is part of the set (inclusive).
• Use parentheses, \(( \text{ and } )\), to indicate that the endpoint is not included in the solution (exclusive).

Thus, \([-3, 2]\) means all numbers from -3 to 2, inclusive of both these values.

Solving Inequalities

Solving inequalities involves determining which values of a variable make the inequality true. The process is slightly different from solving equations, as inequalities describe a range of possible solutions and not just a single value.
To solve an inequality involving absolute value, we break it into more manageable parts. For example, for the inequality \(|2x + 1| \leq 5\), it is split into two separate inequalities:
1. First, solving \(2x + 1 \leq 5\), we subtract 1 and then divide by 2, resulting in \(x \leq 2\).
2. Secondly, solving \(2x + 1 \geq -5\), applying the same steps gives us \(x \geq -3\).
The solution to the original inequality is found by identifying the intersection of these ranges: all numbers for which both conditions hold true, which here is \(-3 \leq x \leq 2\).
Keep in mind, multiplying or dividing an inequality by a negative number flips the inequality sign, a key difference from equality solutions.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In solving the inequality problem, algebraic manipulation was used to isolate variables and simplify expressions.
Key steps involve operations like adding, subtracting, multiplying, or dividing both sides of an inequality while keeping it balanced. For the given exercise:
- First, we subtracted 1 from each side of the inequality to isolate the absolute value part.
- Next, we divided by 2 to solve both the split inequalities derived from the absolute value.
These steps, fundamental in algebra, ensure the correct handling of inequalities and establish the groundwork for finding accurate solutions.