Problem 25 - Algebra Form and Function

Question

For what values of \(x\) are the quantities negative? $$ 3 x+18 $$

Step-by-Step Solution

Answer

Answer: The expression \(3x + 18\) is negative when \(x < -6\).

Step 1: Write down the inequality

Rewrite the expression as an inequality, with the expression being less than zero: $$ 3x + 18 < 0 $$

Step 2: Solve the inequality for \(x\)

In order to solve the inequality, isolate \(x\) on one side of the inequality: $$ 3x < -18 $$ Now, divide by \(3\) on both sides to solve for \(x\): $$ x < -6 $$

Step 3: State the solution

The inequality \(x<-6\) represents the range of values for \(x\) for which the expression \(3x+18\) is negative. So, the given expression is negative when \(x\) is less than \(-6\).

Key Concepts

Solving Inequalities

An inequality is a mathematical statement that shows the order or relation between two values. When solving inequalities, the goal is to find the set of values that satisfy the condition specified by the inequality sign. In the exercise provided, the inequality \( 3x + 18 < 0 \) is to be solved.

To start, we isolate the variable on one side of the inequality. This involves subtracting or adding terms, just like you would when solving an equation. In this case, we subtract 18 from both sides to obtain \( 3x < -18 \).

• Subtracting 18: balances the inequality without changing its nature.
• Dividing both sides by 3: simplifies the inequality to \( x < -6 \).

This final step relies on the principle that inequality signs remain unchanged when multiplying or dividing with positive numbers. Hence, solving the inequality reveals that for \( x < -6 \), the original expression is negative.

Negative Quantities

In mathematics, understanding negative quantities is crucial for interpreting inequality solutions. A negative quantity results when a value is less than zero. In the given problem, the expression \( 3x + 18 \) is to be negative.

This involves setting up the condition \( 3x + 18 < 0 \), capturing that we seek \( x \) values making the whole expression look like a negative number such as \(-1, -10\), or less. Recognizing this allows us to directly apply arithmetic operations, bringing us closer to solving for \( x \).

• Concept: A negative sum can be achieved if the product of two positive numbers is subtracted by something larger.
• Real-life analogy: More bills than cash result in a negative bank account balance.

By solving, as shown in the steps, \( x < -6 \) means if \( x \) is any number less than \(-6\), it makes \( 3x + 18 \) negative.

Linear Inequalities

Linear inequalities are inequalities that involve a linear expression. Linear means the highest power of the variable is one. An example, like \( 3x + 18 < 0 \), indicates the inequality is linear.

Linear inequalities are solved using similar methods as linear equations but with mindful attention to the inequality signs. When multiplying or dividing through with negative numbers, the inequality sign flips direction, a distinction from equations.

• Sign Flip: Applies when multiplying/dividing by a negative.
• Graphing: Solutions often shown on number lines.

In this exercise, actions like subtracting 18 and dividing by 3 kept the inequality direction steady since all interactions were with positive numbers. Thus, the simple linear process ultimately discloses that \( x < -6 \) retains the negative characteristic of the expression \( 3x + 18 \). Linear inequalities, thus, reinforce understanding through clear procedural steps, focusing on numerical relationships and not complex operations.