Problem 21 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ |2 x|>7 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -\frac{7}{2}) \cup (\frac{7}{2}, \infty)\).

Step 1: Understand the Absolute Value Inequality

The inequality given is \(|2x| > 7\). The absolute value inequality \(|a| > b\) means that \(a > b\) or \(a < -b\). This is because the absolute value of a number represents its distance from zero and will be greater than a positive number if it's either greater than that number or less than its negative counterpart.

Step 2: Split the Inequality into Two Cases

Since \(|2x| > 7\), we split it into two inequalities: 1) \(2x > 7\) 2) \(2x < -7\)

Step 3: Solve First Inequality

Solve the inequality \(2x > 7\) by dividing both sides by 2:\[ x > \frac{7}{2} \]

Step 4: Solve Second Inequality

Solve the inequality \(2x < -7\) by dividing both sides by 2:\[ x < -\frac{7}{2} \]

Step 5: Express Solution in Interval Notation

The solutions \(x > \frac{7}{2}\) or \(x < -\frac{7}{2}\) can be expressed in interval notation as \((-\infty, -\frac{7}{2}) \cup (\frac{7}{2}, \infty)\).

Step 6: Verify the Solution

Verify the solution by considering values from each interval:- For \(x < -\frac{7}{2}\), choose \(x = -4\). Check: \(|2(-4)| = 8 > 7\). True.- For \(x > \frac{7}{2}\), choose \(x = 4\). Check: \(|2(4)| = 8 > 7\). True.The interval notation matches the correct solutions.

Key Concepts

Absolute Value Inequality

Understanding absolute value inequalities is crucial in solving problems like \(|2x| > 7\). The absolute value of a number represents its distance from zero on a number line. When dealing with an inequality such as \(|a| > b\), this implies that:

• The expression inside the absolute value, \(a\), is greater than \(b\).
• Alternatively, \(a\) is less than \(-b\).

This splitting of the inequality into two parts is due to the nature of absolute values which can result in equivalent positive or negative numbers. So, for \(|2x| > 7\), it translates to two separate conditions: \(2x > 7\) and \(2x < -7\). This approach helps to consider all possible values of \(x\) that satisfy the condition, fully encompassing the range of solutions entailed by absolute measures.

Interval Notation

Interval notation is a way of expressing solutions to inequalities clearly and comprehensively. In our example, the solution to \(|2x| > 7\) yielded two separate inequalities: \(x > \frac{7}{2}\) and \(x < -\frac{7}{2}\).
Using interval notation, we represent:

• \(x < -\frac{7}{2}\) as \((-\infty, -\frac{7}{2})\)
• \(x > \frac{7}{2}\) as \((\frac{7}{2}, \infty)\)

These intervals exclude the endpoints because the absolute inequality \(|2x| > 7\) does not include equality to 7 or -7. We combine these intervals using the union symbol (\(\cup\)) to indicate that \(x\) can belong to either interval. Thus, the expression \((-\infty, -\frac{7}{2}) \cup (\frac{7}{2}, \infty)\) fully depicts the range of \(x\) where the inequality is satisfied.

Solving Inequalities

When solving inequalities, like those found in absolute value problems, it involves isolating the variable to determine the set of possible solutions. This process often requires splitting the problem into different segments, as seen with \(|2x| > 7\).
Here's how it unfolds:

• For \(2x > 7\), simply divide both sides by 2 to solve for \(x\). This yields \(x > \frac{7}{2}\).
• For \(2x < -7\), again divide both sides by 2. This results in \(x < -\frac{7}{2}\).

By resolving each segment separately, you build a comprehensive understanding of where the inequality holds. Always remember to verify your solutions by testing points within the determined intervals. This ensures accuracy and sound understanding of the concepts involved in solving such inequalities.