Problem 67 - College Algebra

Question

Solve absolute value inequality. \(\left|\frac{2 x+6}{3}\right|<2\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x \in (-6, 0)\).

Step 1: Express the inequality without the absolute value

Express the inequality into two separate inequalities, because the value inside the absolute value can either be negative or positive without changing the final value. This gives us the two inequalities: \( \frac{2x + 6}{3} < 2 \) and \( \frac{2x + 6}{3} > -2 \)

Step 2: Solve the first inequality

Solve \( \frac{2x + 6}{3} < 2 \) by first multiplying both sides by 3 to get rid of the fraction: \( 2x + 6 < 6 \). Then, subtract 6 from both sides: \(2x < 0\).Finally, divide by 2: \(x < 0\).

Step 3: Solve the second inequality

Solve \( \frac{2x + 6}{3} > -2 \) by first multiplying both sides by 3: \( 2x + 6 > -6 \). Then, subtract 6 from both sides: \(2x > -12\).Finally, divide by 2: \(x > -6\).

Step 4: Intersection of the solutions

Since the original inequality had an 'and' condition (meaning both inequalities must hold true), the final solution is the intersection of the individual solutions. In other words, only those x values that satisfy both conditions are valid solutions. The intersection of \(x < 0\) and \(x > -6\) gives us the final solution: \(x \in (-6, 0)\).

Key Concepts

Inequality Solving

Solving inequalities involves finding all values of the variable that make the given inequality true.
Absolute value inequalities, like the one in the problem, add another layer of complexity. The absolute value, \( | \, \cdot \, | \), represents the distance from zero and is always non-negative.
When the absolute value is in an inequality, it divides the problem into two separate inequalities: one where the expression inside the absolute value is positive and the other where it is negative.

• For \( |X| < a\), the inequality is split into \(X < a\) and \(X > -a\).
• Both conditions must be satisfied simultaneously, leading to the need for an 'and' condition between them.

Both new inequalities are solved separately, and their solutions are combined to form the final solution.

Algebraic Expressions

Algebraic expressions are key to solving inequalities effectively. An algebraic expression, in this context, involves a combination of variables, numbers, and operations like addition and multiplication.
In the problem, the expression \(\frac{2x + 6}{3}\) represents the entire term inside the absolute value brackets.
When dealing with such expressions, it’s crucial to:

• Use operations strategically to simplify the expression. Here, multiplying by 3 removes the fraction.
• Perform operations step-by-step, such as subtraction and division, to isolate the variable \(x\).

These steps reduce the complexity of the expressions and bring us closer to finding the solution to the inequality. Solving equations within inequalities involves manipulating these expressions while maintaining balance on both sides of the inequality sign.

Solution Sets

The solution set of an inequality is simply all the values of the variable that satisfy the inequality. In the case of the absolute value inequality given, the task was to determine which values of \(x\) work in both resulting inequalities.
By solving each one, we get: \(x < 0\) from the first inequality and \(x > -6\) from the second.

• The solution must satisfy both conditions due to the nature of 'and' combined inequalities.
• Therefore, the solution set is the overlap, or intersection, of these two conditions.

For the given example, that means the values of \(x\) are between \( -6 \) and \( 0\), written as the interval \(x \in (-6, 0)\).
This interval notation concisely shows every possible solution to the inequality.