Problem 35 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ 4|x+2|-3<13 $$

Step-by-Step Solution

Answer

The solution is x ∈ (-6, 2).

Step 1: Isolate the Absolute Value Expression

Start by isolating the absolute value on one side of the inequality. Add 3 to both sides to get: \[4|x+2| < 16 \]Now, divide both sides by 4 to further isolate the absolute value:\[|x+2| < 4\]

Step 2: Set Up Two Inequalities

The expression \(|x+2| < 4\) can be split into two separate inequalities. This is because the absolute value being less than a number means that the expression inside the absolute value must be between the negatives and positives of that number:\[-4 < x+2 < 4\]

Step 3: Solve Each Inequality

Solve the inequalities individually. Start with the first one:\[-4 < x+2\]Subtract 2 from both sides:\[-6 < x\] Now solve the second inequality:\[x+2 < 4\]Subtract 2 from both sides:\[x < 2\]

Step 4: Combine the Solutions

The solutions are in the form of a double inequality:\[-6 < x < 2\]This means that the solution set is all values between -6 and 2, not including -6 and 2.

Step 5: Express the Solution in Interval Notation

Translate the double inequality into interval notation. Since the inequality signs are strict (not \( \leq \)), the interval does not include the endpoints:\[(-6, 2)\]

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve an expression within absolute value symbols that is related to a constant number through an inequality sign. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, when solving absolute value inequalities, the core idea is to understand how far the expression inside the absolute value could be from zero but still satisfy the given condition.

Consider an inequality like \(|x| < a\)\, where \(a\) is a positive number. This inequality implies that \(x\) is within \(-a\) and \(a\). In terms of solving techniques, we split the inequality into two separate inequalities: \-a < x < a\. Solving each provides a range of values that make the original absolute value inequality true.

Once solved, these expressions can be combined into a double inequality for simplicity and easy interpretation, representing a range of permissible values within specified bounds.

Interval Notation

Interval notation is a concise way of expressing the set of numbers that belong to a specific range. It uses parentheses and brackets to denote open and closed intervals, respectively.

- Parentheses \(( )\) signify that the interval does not include the endpoint, corresponding to the "less than" or "greater than" inequalities (\(<\) or \(>\)).
- Brackets \([ ]\) imply that the interval does include the endpoint, which aligns with "less than or equal to" or "greater than or equal to" inequalities (\(\leq\) or \(\geq\)).

For example, in the original solution, the inequality \-6 < x < 2\ translates to the interval \((-6, 2)\). This interval representation indicates that \(x\) can take any value between \(-6\) and \(2\), excluding the endpoints since the inequality is strict.

Double Inequalities

Double inequalities are powerful and compact notations that describe a range of values between two boundaries. They are especially handy when dealing with absolute value inequalities because they seamlessly bridge two related inequalities into one neat expression.

In a typical double inequality like \-4 < x+2 < 4\, both inequalities must hold true simultaneously. You effectively solve each part separately to find a common solution. The left part \-4 < x+2\ indicates values greater than \-4\, while the part \x+2 < 4\ suggests values less than \(4\). Combining these results in a solution that governs the full behavior of the inequality.

Double inequalities are read left to right, maintaining the order between the smallest and largest values. In our example, solving and consolidating into a single \(-6 < x < 2\) informs us where the solution lies on the number line.