Problem 80 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ 5|2 x+1|-3 \geq 9 $$

Step-by-Step Solution

Answer

The solution set for the given absolute value inequality is \(-\frac{17}{10} \leq x \leq \frac{7}{10}\).

Step 1: Isolate the Absolute Value Expression

The first step is to isolate the absolute value expression. We'll start with the original inequality \(5|2x+1|-3 \geq 9\) and isolate the absolute value by adding 3 to both sides, and then divide everything by 5. That gives us: \(|2x+1| \geq \frac{12}{5}\). Now we have the absolute value expression alone on one side.

Step 2: Express the Inequality Without the Absolute Value

The fact that the absolute value of a number is greater than or equal to another number means that the number is either greater than or equal to the positive value of that number, or is less than or equal to the negative value of that number. Therefore, we can rewrite the inequality without the absolute value as a compound inequality: \(2x+1 \geq \frac{12}{5}\) or \(2x+1 \leq - \frac{12}{5}\).

Step 3: Solve the Compound Inequality

In this step, we will solve both inequalities. To do this, we subtract 1 from both sides and then divide by 2 to solve for x. For the first inequality, \(\frac{2x+1-1}{2} \geq \frac{12}{5}-1\), we get \(x \geq \frac{7}{10}\). For the second inequality, \(\frac{2x+1-1}{2} \leq -\frac{12}{5}-1\), we get \(x \leq -\frac{17}{10}\). The solution set for this inequality is \(-\frac{17}{10} \leq x \leq \frac{7}{10}\).

Key Concepts

Solving Inequalities

Understanding how to solve inequalities is a fundamental skill in algebra. An inequality, unlike an equation, indicates that one side is not necessarily equal to the other, but rather greater than, less than, or possibly equal when we're dealing with non-strict inequalities. When we solve inequalities, it’s about finding all the possible values that make the inequality true.

General Steps for Solving Inequalities

• Identify the Inequality: Determine which type of inequality you are dealing with (greater than, less than, greater than or equal to, or less than or equal to).
• Perform Operations: Apply operations to both sides of the inequality to isolate the variable just like you would with an equation, being careful to reverse the inequality sign when multiplying or dividing by a negative number.
• Check the Solution: Unlike equations, where you'd plug the solution back in to check, with inequalities, it's often more practical to test a range of values around your solution to ensure the inequality holds true.
• Express the Solution: When the solution is a range of numbers, it's commonly expressed with inequality notation, a graph, or interval notation.

The exercise provided involves an inequality with an absolute value, which adds a layer of complexity as we have to consider both the positive and negative scenarios for the expression within the absolute value.

Compound Inequality

A compound inequality is an inequality that combines two distinct inequalities into one statement by using the word 'and' or 'or'. When the word 'and' is used, it means that the solution must satisfy both inequalities at once. On the other hand, 'or' implies that either of the inequalities can be true for the solution to hold.

In the context of absolute value inequalities, creating a compound inequality is often necessary. For an expression like \(|2x+1| \geq \frac{12}{5}\), the absolute value creates two scenarios: when \(2x+1\) is positive and when it's negative. Therefore, we split it into two different inequalities (\(2x+1 \geq \frac{12}{5}\) or \(2x+1 \leq -\frac{12}{5}\)) and treat them as a compound inequality.

Visualizing Compound Inequalities

If you graph the solutions of these inequalities on a number line, 'and' would be where the solution sets intersect, whereas 'or' would be the union of both solutions sets, depicting all the numbers that make either of the inequalities true.

Isolate Absolute Value Expression

When solving absolute value inequalities, a critical step is to isolate the absolute value expression on one side of the inequality. This technique is important because it allows us to treat the absolute value expression as a 'single entity', which we can then analyze and split into its positive and negative components.

To isolate the absolute value, similar to isolating a variable, we perform inverse operations to 'undo' anything around the absolute value. This often involves adding or subtracting terms on both sides and dividing or multiplying to get the absolute value by itself, as is done in the textbook example with the expression \(5|2x+1|-3 \geq 9\).

Once we have the absolute value expression isolated, such as \(|2x+1| \geq \frac{12}{5}\), it becomes clear to define the two scenarios under which the inequality will hold true. Isolating the absolute value is a key step that sets up the equation for the creation of a compound inequality, leading us towards finding the full range of the solution.