Problem 16 - College Algebra

Question

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation. $$ |2 x+3| < 7 $$

Step-by-Step Solution

Answer

The solution is \((-5, 2)\).

Step 1: Understand the Absolute Value Inequality

The inequality \(|2x + 3| < 7\) states that the expression inside the absolute value must be less than 7 away from 0. This implies two separate inequalities: \(2x + 3 < 7\) and \(2x + 3 > -7\).

Step 2: Solve the First Inequality

Consider the inequality \(2x + 3 < 7\).Subtract 3 from both sides: \(2x < 4\).Then divide both sides by 2: \(x < 2\).

Step 3: Solve the Second Inequality

Consider the inequality \(2x + 3 > -7\).Subtract 3 from both sides: \(2x > -10\).Then divide both sides by 2: \(x > -5\).

Step 4: Combine Solutions and Express in Interval Notation

Combine the inequalities from Steps 2 and 3: \(-5 < x < 2\).This is expressed in interval notation as \((-5, 2)\).

Key Concepts

Interval Notation

Interval notation is a mathematical way to represent a set of solutions to inequalities, such as \(-5 < x < 2\).It clearly conveys which numbers are part of the solution set. Interval notation uses brackets and parentheses:

• Parentheses \((a, b)\) indicate that \(a\) and \(b\) are not included (open interval).
• Brackets \([a, b]\) indicate that \(a\) and \(b\) are included (closed interval).

For the inequality \(-5 < x < 2\), it means \(x\) can be any number between \(-5\) and \(2\),not including \(-5\) or \(2\).Thus, the interval notation is \((-5, 2)\).This concise notation efficiently communicates the valid range of values for the given inequality.

Solving Inequalities

Solving inequalities involves finding which values of a variable satisfy the inequality condition. For absolute value inequalities like \(|2x + 3| < 7\), we interpret the inequality to form two separate inequalities.

• First, solve \(2x + 3 < 7\): Subtract \(3\) from both sides, giving \(2x < 4\). Divide by \(2\) to find \(x < 2\).
• Next, solve \(2x + 3 > -7\): Again, subtract \(3\) from both sides, yielding \(2x > -10\). Divide by \(2\) to get \(x > -5\).

Once you solve both, combine the solutions to form a range: \(-5 < x < 2\).This means any \(x\) within that range satisfies the original inequality,excluding \(-5\) and \(2\) from the solution.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. In our exercise, \(2x + 3\) is the algebraic expression we need to handle within the absolute value inequality.

• The term \(2x\) represents a multiplication operation, indicating that \(x\) is scaled by a factor of two.
• The \(+3\) part signifies a shift upwards by three units on the number line.

Understanding these expressions is essential for solving inequalities efficiently,as they help determine how to manipulate and balance equations.Remember that changes made to one side of an inequality must likewise apply to the other side,to maintain equality or inequality relationships consistently.