Problem 19 - Algebra Form and Function

Question

Solve the inequality. $$ |3 z-9|>4 $$

Step-by-Step Solution

Answer

Question: Solve the inequality \(|3z - 9| > 4\). Answer: The inequality is true when \(z < \frac{5}{3}\) or when \(z > \frac{13}{3}\).

Step 1: Break down the absolute value inequality into two cases

Since the expression inside the absolute value is either positive or negative, we can write the inequality as two separate inequalities. $$ 3z - 9 > 4 $$ and $$ -(3z - 9) > 4 $$

Step 2: Solve the first inequality

Let's now solve the inequality \(3z - 9 > 4\). $$ 3z - 9 > 4 \Rightarrow 3z > 13 \Rightarrow z > \frac{13}{3} $$ So, when \(3z -9\) is positive, we get that \(z > \frac{13}{3}\).

Step 3: Solve the second inequality

Next, let's solve the inequality \(-(3z - 9) > 4\). $$ -(3z - 9) > 4 \Rightarrow -3z + 9 > 4 \Rightarrow -3z > -5 \Rightarrow z < \frac{5}{3} $$ So, when \(3z - 9\) is negative, we get that \(z < \frac{5}{3}\).

Step 4: Combine the solutions

Now we combine both inequalities to present the final solution to the problem. The solutions are: $$ z < \frac{5}{3} $$ and $$ z > \frac{13}{3} $$ This means that the inequality \(|3z - 9| > 4\) is true when \(z < \frac{5}{3}\) or when \(z > \frac{13}{3}\).

Key Concepts

Absolute Value

The concept of absolute value is crucial when dealing with inequalities. Simply put, the absolute value of a number is its distance from zero on the number line, without considering which direction. For instance, both 3 and -3 have an absolute value of 3 because they are both three units away from zero.

When we have an absolute value inequality like \(|3z - 9| > 4\), it tells us the distance between \(3z - 9\) and zero is greater than 4. This information is vital as it clues us into how we will set up our problem for solving. By removing the absolute value, we can consider two cases: when the expression is positive and when it is negative.

• If the inside expression \((3z - 9)\) is greater than 4, it's already clear that we exceed the distance from zero by being positively > 4.
• If \(-(3z - 9) > 4\), it represents that the negative of the expression is positive, meaning without its sign it's also more than 4 units away from zero.

Understanding these premises allows us to break down absolute value inequalities into linear equations. This division into two distinct inequalities is key in finding solutions.

Algebraic Manipulation

Algebraic manipulation is about changing the form of an inequality to find the solution. For the inequality \(3z - 9 > 4\), we start by isolating the term with the variable "z".

• Add 9 to both sides: \(3z > 13\).
• Divide each side by 3: \(z > \frac{13}{3}\).

This gives us the solution for when the expression inside the absolute value is positive.

For the negative scenario \(-(3z - 9) > 4\), we start by distributing the negative sign:

• Simplify to \(-3z + 9 > 4\).
• Subtract 9 from both sides: \(-3z > -5\).
• Divide each side by -3, and remember that dividing by a negative flips the inequality: \(z < \frac{5}{3}\).

These steps help in getting a clear solution while ensuring that you maintain balance and correctness throughout the manipulation process.

Linear Inequalities

Linear inequalities are expressions that involve a linear expression in relation to another expression. In the context of the given problem, we deal with linear expressions as they involve the variable "z" raised only to the first power.

To solve these inequalities, follow these general steps:

• Reform the absolute value into two linear inequalities.
• Use simple algebraic manipulation to solve each inequality.
• Remember that dividing or multiplying by a negative number flips the inequality sign.
• Combine your results to understand the solution set thoroughly.

In this exercise, the final solution after solving the expressions is \(z < \frac{5}{3}\) or \(z > \frac{13}{3}\).

The inequality is true for values of "z" outside the interval \((\frac{5}{3}, \frac{13}{3})\). This solution means that these are the values for which the original inequality holds true. Linear inequalities like this can sometimes be mistaken to only hold values within a range, but absolute values require us to consider values outside. Recognizing this interaction is quintessential in solving such problems.