Problem 77 - College Algebra

Question

Solve absolute value inequality. \(\left|3-\frac{2}{3} x\right|>5\)

Step-by-Step Solution

Answer

The solution to the inequality \(\left|3-\frac{2}{3} x\right|>5\) is \(x < -3\) or \(x > 12\).

Step 1: Rewrite as Two Separate Inequalities

The initial inequality is \(\left|3-\frac{2}{3} x\right|>5\). An absolute value of a quantity being greater than something means that the quantity can be less than the negative of that something or greater than that something. This translates into two separate inequalities: \(3-\frac{2}{3} x > 5\) and \(3-\frac{2}{3} x < -5\).

Step 2: Solve Each Inequality

Now, solve each inequality individually. In solving \(3-\frac{2}{3} x > 5\), isolate x to obtain \(\frac{2}{3}x < 3 - 5\), which simplifies to \(\frac{2}{3}x < -2\). Multiplying each side by \(\frac{3}{2}\) (remembering to flip the inequality sign because you are dividing by a negative number) results in \(x < -3\). When solving \(3-\frac{2}{3} x < -5\), isolate \(x\) to get \(\frac{2}{3}x > 3 + 5\), which simplifies to \(\frac{2}{3}x > 8\). Multiplying each side by \(\frac{3}{2}\) results in \(x > 12\).

Step 3: Combine the Results

From Step 2, we found that \(x < -3\) or \(x > 12\). This is the solution set for the original absolute value inequality.

Key Concepts

Solving Inequalities

Inequalities are similar to equations, but instead of using an equal sign, they use inequality symbols such as \( <, \leq, >, \geq \). Solving an inequality often involves finding the set of values that makes the inequality true. Just like solving equations, you can add, subtract, multiply, and divide both sides of an inequality by the same number. However, there's a crucial difference: when you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This ensures that the inequality remains valid.Let's break down the process:

• Identify the inequality and write it clearly.
• Isolate the variable of interest by rearranging the inequality just like you would in a regular equation.
• Perform any necessary operations to simplify the inequality.
• Pay special attention when multiplying or dividing by negative numbers. Remember to flip the inequality sign.

By following these steps, you can confidently solve inequalities and find the range of possible solutions that satisfy them.

Absolute Value Functions

Absolute value is a mathematical function that measures how far a number is from zero, regardless of its sign. The absolute value of a number \( x \) is denoted by \( |x| \) and is always a non-negative value.This concept is useful in real-world contexts where you're interested in the magnitude of change or distance without regard for the direction. For absolute value inequalities, we deal with expressions like \( |a| > b \), \( |a| < b \), \( |a| \leq b \), and \( |a| \geq b \).To solve absolute value inequalities:

• Translate the absolute value expression into two separate inequalities.
• If \( |a| > b \), then \( a > b \) or \( a < -b \).
• If \( |a| < b \), then \( -b < a < b \).

By understanding how absolute values are interpreted in inequalities, you can tackle questions with ease and assurance.

Compound Inequalities

Compound inequalities involve more than one inequality linked together by the word 'and' (\( \land \)) or 'or' (\( \lor \)). These expressions help in defining a range or multiple ranges of possible solutions.Compound inequalities can be very useful in stating ranges within which a solution must fall. For instance, in the exercise we analyzed, we derived two separate inequalities: \( x < -3 \) or \( x > 12 \), which means that \( x \) can be in two distinct ranges. When solving compound inequalities involving 'or':

• Find the solution to each inequality separately.
• The final solution is the union of sets found from each inequality. This means that any number that satisfies one or more of the individual inequalities is a solution to the compound inequality.

In compound inequalities involving 'and', the solution must satisfy both inequalities simultaneously, thereby limiting the range more strictly.These concepts are foundational for understanding how multiple conditions can simultaneously affect the range of solutions for a mathematical problem.