Problem 77 - Algebra and Trigonometry

Question

In Exercises 59–94, solve each absolute value inequality. $$ 3-\frac{2}{3} x |>5 $$

Step-by-Step Solution

Answer

The solutions for the inequality are \(x < -3\) and \(x > 3\).

Step 1. Isolate absolute value expression.

Subtract 3 from both sides of the inequality to isolate the absolute value: \(\|- \frac{2}{3}x\| > 2\).

Step 2. Split inequality into two

\(\frac{2}{3}x < -2\) which can be written as \(x < -3\) and \(\frac{2}{3}x > 2\) which can be written as \(x > 3\). This is done because of the absolute value's property - it pulls 'x' to either side of 0.

Step 3: Final answers

The solution to the inequality \(\|3 - \frac{2}{3}x\| > 5\) is \(x < -3\) or \(x > 3\).

Key Concepts

Isolation of Absolute Value

When tackling absolute value inequalities, the first step is to isolate the absolute value expression. In simple terms, this means that you want the absolute value part by itself on one side of the inequality. This isolation makes it easier to deal with the inequality without extra distractions. Consider the original problem: \( 3 - \frac{2}{3}x > 5 \). To isolate the absolute value \( |-\frac{2}{3}x | \), you subtract 3 from both sides of the inequality. After subtraction: - The equation becomes \( |-\frac{2}{3}x| > 2 \).At this stage, you have the absolute value expression on one side and a real number on the other. This sets the stage for the next steps of solving the inequality. Being adept at isolating the absolute value will help tremendously with these types of problems.

Inequality Splitting

Once you have isolated the absolute value, it's time to split the inequality. Absolute value can unfold into two possibilities because it represents distance from zero on a number line. Each absolute value inequality can be split into two separate inequalities.For \( |-\frac{2}{3}x | > 2 \), we engage in splitting to address both scenarios:- \(-\frac{2}{3}x < -2\)- \(-\frac{2}{3}x > 2\)The first inequality, \(-\frac{2}{3}x < -2\), resolves to \(x < -3\) after reversing signs and solving for \(x\).The second inequality, \(-\frac{2}{3}x > 2\), resolves to \(x > 3\) in a similar manner.Remember, the reason for creating two inequalities stems from the property of absolute value—the expression can represent both positive and negative scenarios. This crucial step ensures that you're considering all possible solutions in the equation.

Solving Inequalities

The final and essential task is solving the split inequalities. This step involves working through basic algebraic manipulations to find the values of \(x\) that satisfy each inequality.From the previously split statements:- \(x < -3\)- \(x > 3\)These inequalities tell us about the range of values that \(x\) can take. The solution to the problem, therefore, is not restricted to a single set of numbers but split into two main outcomes:

• If \(x < -3\), the distance from zero remains greater than 5.
• Similarly, if \(x > 3\), the distance from zero is also greater than 5.

Thus, the union of these inequalities gives the complete solution. In simpler terms, \(x\) can lie anywhere outside the interval \([-3, 3]\), effectively meaning \(x\) is less than \(-3\) or greater than \(3\). These insights resolve the inequality \(|-\frac{2}{3}x| > 2\), helping complete the solution with \(x < -3\) or \(x > 3\).