Problem 25 - Algebra and Trigonometry

Question

\(23-48\) Solve the inequality. Express the answer using interval notation. $$ |2 x|>7 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Inequality

The given inequality is \(|2x| > 7\). The absolute value inequality \(|a| > b\) can be rewritten as two separate inequalities: \(a > b\) or \(a < -b\).

Step 2: Separate the Inequality

For \(|2x| > 7\), rewrite as two inequalities: 1. \(2x > 7\) 2. \(2x < -7\).

Step 3: Solve the First Inequality

Solve \(2x > 7\). Divide both sides by 2:\[x > \frac{7}{2}\].

Step 4: Solve the Second Inequality

Solve \(2x < -7\). Divide both sides by 2:\[x < -\frac{7}{2}\].

Step 5: Express in Interval Notation

Combine the solutions. The solution for the inequality \(|2x| > 7\) is:\(x > \frac{7}{2}\) or \(x < -\frac{7}{2}\).In interval notation, this is expressed as:\((-\infty, -\frac{7}{2}) \cup (\frac{7}{2}, \infty)\).

Key Concepts

Inequality Solutions

Inequality solutions involve finding the range of values that make an inequality true. In the given problem, we were tasked with solving the inequality \(|2x| > 7\).
The absolute value symbol typically indicates that the expressions inside it can take on both positive and negative values. This leads us to separate it into two inequalities:

• \(2x > 7\)
• \(2x < -7\)

Each inequality must be solved individually to determine two separate ranges of \(x\) values that are valid solutions. Separating absolute value inequalities is crucial because the solution set often comprises two distinct parts.

Interval Notation

Once both inequalities are solved, you'll need to express your solutions using interval notation. Interval notation succinctly communicates the solution set for the inequality using brackets:

• \(([a, b])\) where \(a\) and \(b\) are included in the interval.
• \((a, b)\) where \(a\) and \(b\) are not included.
• \((-\infty, b)\) or \((a, \infty)\), where infinity is always written with a parenthesis.

For our particular solution, the inequality \(|2x| > 7\) breaks into two parts, resulting in the intervals \((-\infty, -\frac{7}{2})\) and \((\frac{7}{2}, \infty)\). We use the union symbol \(\cup\) to indicate that \(x\) can be in either part, providing a complete picture of all possible values of \(x\) that satisfy the original inequality.

Solving Inequalities

Turning an inequality into a solved statement means isolating the variable on one side. For absolute values, we first rewrite the inequality as two separate inequalities, as shown previously. Each one will be solved similarly to regular linear equations.

• For \(2x > 7\), divide both sides by 2 to isolate \(x\), resulting in \(x > \frac{7}{2}\).
• For \(2x < -7\), again divide by 2, leading to \(x < -\frac{7}{2}\).

These procedures help us find the individual solution sets for the variable \(x\). Solving inequalities requires careful steps to maintain the inequality’s direction, especially when multiplying or dividing by negative numbers.

Mathematical Inequalities

Mathematical inequalities are expressions showing that one quantity is larger or smaller than another. They use symbols like \(>\), \(<\), \(\geq\), and \(\leq\). Inequalities can express a broad range of real-world situations such as comparing values, limits, or thresholds.
By solving the inequality \(|2x| > 7\), we determine intervals that satisfy the condition that the absolute value of \(2x\) remains greater than 7. Frequently, inequalities require careful manipulation and interpretation, especially when absolute values are involved, because they represent two potential scenarios — often leading to a union of sets in the solution.
Understanding how mathematical inequalities function allows for their application in various mathematical and practical contexts, enhancing problem-solving skills across diverse areas.