Problem 66 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ \frac{2}{|2 x+3|} \geq 5 $$

Step-by-Step Solution

Answer

The solution is \([-\frac{17}{10}, -\frac{13}{10}]\).

Step 1: Understand the Problem

We need to solve the inequality \( \frac{2}{|2x+3|} \geq 5 \). This is an inequality involving an absolute value. Absolute values make the expression \(|2x+3|\) always non-negative.

Step 2: Isolate the Absolute Value

Multiply both sides by \(|2x+3|\) to eliminate the fraction, getting \(2 \geq 5|2x+3|\). Make sure \(|2x+3| eq 0\) so this step does not violate any properties.

Step 3: Divide to Simplify

Divide both sides by 5, resulting in \(|2x+3| \leq \frac{2}{5}\). Now we have a standard absolute value inequality.

Step 4: Solve the Absolute Value Inequality

The inequality \(|2x+3| \leq \frac{2}{5}\) implies two separate inequalities: \(-\frac{2}{5} \leq 2x+3 \leq \frac{2}{5}\).

Step 5: Solve Both Inequalities

First solve \(2x+3 \leq \frac{2}{5}\) by subtracting 3: \(2x \leq \frac{2}{5} - 3\). Simplify to get \(2x \leq -\frac{13}{5}\). Divide by 2 to find \(x \leq -\frac{13}{10}\).

Step 6: Solve the Other Inequality

Now solve \(-\frac{2}{5} \leq 2x+3\) by subtracting 3: \(-\frac{2}{5} - 3 \leq 2x\). Simplify to get \(-\frac{17}{5} \leq 2x\). Divide by 2 to find \(-\frac{17}{10} \leq x\).

Step 7: Combine the Solutions

Combine the solutions from steps 5 and 6: \(-\frac{17}{10} \leq x \leq -\frac{13}{10}\). Express this as an interval: \([-\frac{17}{10}, -\frac{13}{10}]\).

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It is often used to represent solution sets for inequalities. Since interval notation succinctly shows the beginning and end of a set, it's both efficient and clear.
When writing in interval notation, we use brackets depending on whether the endpoint is included:

• Square brackets "[" or "]" indicate that the endpoint is included in the set.
• Parentheses "(" or ")" indicate that the endpoint is not included in the set.

When we solved the inequality \([-\frac{17}{10}, -\frac{13}{10}]\), both numbers are included in the solution set, thus square brackets are used. This tells us that our variable, \(x\), includes all values from \(-\frac{17}{10}\) to \(-\frac{13}{10}\).
Interval notation helps make the range of solutions clear at a glance, making it easier for you to understand which numbers fit an inequality.

Solving Inequalities

Solving inequalities is similar to solving equations but with a few additional rules that need to be followed. For instance, when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol.
A step-by-step approach is often helpful:

• Isolate the variable you're solving for.
• Perform algebraic operations step by step.
• Be mindful of reversing the sign when necessary.

In the example inequality \(2\), after isolating \(|2x+3|\), we derived simpler inequalities by breaking down the absolute value expression into two separate, linear inequalities. This approach simplifies the process and prevents errors.

Inequalities with Absolute Values

Inequalities involving absolute values require special attention because they express distance from zero. For an inequality like \(|2x+3| \leq \frac{2}{5}\), the expression indicates that \(2x+3\) lies within \(\frac{2}{5}\) units from zero. This means the actual expressions to solve are \(-\frac{2}{5} \leq 2x+3 \leq \frac{2}{5}\).
These are then broken down into two inequalities to solve separately:

• \(2x+3 \leq \frac{2}{5}\)
• \(-\frac{2}{5} \leq 2x+3\)

By solving these, you find the range of solutions for \(x\). The absolute value essentially doubles the work, but once you grasp the core concept, this method is logical and straightforward.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Understanding them is crucial for manipulating equations and inequalities. In this problem, \(|2x+3|\) is an algebraic expression that includes an absolute value.
When working with expressions:

• Simplify wherever possible to make calculations easier.
• Perform operations in a systematic manner to ensure accuracy.

For example, isolating the algebraic expression \(|2x+3|\) was a critical step in simplifying the inequality and eventually led us to the absolute value inequalities. Mastery of algebraic manipulation also means gaining a solid understanding of how to work with absolute values and inequalities in tandem.