Problem 74 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ |5 x-2|>13 $$

Step-by-Step Solution

Answer

The solutions for the given inequality \(|5x - 2| > 13\) are \(x > 3\) or \(x < -2.2\).

Step 1: Address Positive Case

First, deal with the condition where the value inside the absolute value brackets is positive, so solve for: \(5x - 2 > 13\). Here, isolate x by first adding 2 to both sides of the inequality to give: \(5x > 15\). Then, divide the result by 5 to isolate x: \(x > 3\). So, for the case where the inside of the absolute value is positive, \(x > 3\). It's called the positive case because we're considering the situation where the quantity \(5x - 2\) is positive.

Step 2: Address Negative Case

Next, look at the condition where the value inside the absolute value brackets is negative, which means: \(5x - 2 < -13\). Again, isolate x by first adding 2 to both sides to give: \(5x < -11\). Then divide the result by 5 to isolate x: \(x < -2.2\). So, for the case where the inside of the absolute value is negative, \(x < -2.2\). This is called the negative case because we're considering the situation where the quantity \(5x - 2\) is negative.

Step 3: Combine the Results

Finally, combine the results from Step 1 and Step 2. The two results are considered separately because of the absolute value, and both must be true for the original inequality to be true. As such, the solution to the original inequality \(|5x - 2| > 13\) is the union of the solution sets from the positive and negative case: \(x > 3\) or \(x < -2.2\). This reflects the fact that for any number x in this solution set, the original inequality is satisfied.

Key Concepts

Positive Case in Inequalities

When solving absolute value inequalities, the first step is to address the positive case of the expression inside the absolute value. The positive case assumes that the expression is greater than zero.

• For example, consider the inequality \( |5x - 2| > 13 \).
• We begin by assuming the expression \(5x - 2\) is positive.
• This assumption leads to the inequality \(5x - 2 > 13\).

To solve this:- Add 2 to both sides, getting \(5x > 15\).- Divide both sides by 5, resulting in \(x > 3\).
By solving these steps, we find that when the expression inside the absolute value is positive, the solution \(x > 3\) satisfies the inequality. This approach helps isolate one part of the compound absolute value problem.

Negative Case in Inequalities

The negative case considers that the value inside the absolute brackets could be negative. So, we explore the scenario where the expression inside the absolute value is less than zero.

• In our example, this translates to \(5x - 2 < -13\).

The reasoning is: subtracting a larger number keeps it negative.- First, add 2 to both sides, resulting in \(5x < -11\).- Then, divide by 5 to isolate the variable \(x < -2.2\).
This tells us how values of \(x\) where \(5x - 2\) is negative still satisfy the original inequality. Solving the negative case gives insights into situations where the expression is less than zero, allowing us to complete the inequality's puzzle.

Solving Inequalities

Solving inequalities involves addressing conditions where each piece contributes to the whole equation's truth. In this case, two separate inequalities emerge due to the absolute value function.

• The absolute nature \(|5x - 2| > 13\) means separating into both positive and negative cases.
• Each case generates a different inequality to solve.

In precisely solving:- Positive case yielded \(x > 3\).- Negative case yielded \(x < -2.2\).
The act of solving means you respect these conditions for being true simultaneously, ultimately helping in determining special solution sets. Inequalities require this dual attention due to the possible values any absolute scenario might cover.

Solution Sets in Inequalities

When collecting all solutions from each case, you're essentially assembling the solution set for the inequality. Solution sets here highlight regions on a number line where inequality's conditions hold.

• We combine outcomes from both scenarios—positive (\(x > 3\)) and negative cases (\(x < -2.2\)).

Solution sets for absolute inequalities can express these neatly:- The union represents the entire range meeting the absolute condition.- Visualizing this, you'd mark two separate regions on a number line.
Thus, the complete solution set encompasses \(x > 3\) or \(x < -2.2\), showing spaces outside the main conditions still satisfying the broader inequality. These results help frame a picture of values adhering to the initial problem and are critical in understanding inequalities' boundaries.