91Ó°ÊÓ

Linear functions are a fundamental concept in algebra. They have the general form:
\[ f(x) = mx + b \]where

• \( m \) represents the slope of the line
• \( b \) is the y-intercept

A linear function creates a straight line when graphed.
The slope \( m \) influences how steep the line is, and the y-intercept \( b \) tells us where the line crosses the y-axis.
In the given exercise, the functions are: \[ f(x) = 8x - 9 \] and \[ g(x) = 3x - 11 \]Both of these functions follow the form \( f(x) = mx + b \) and \( g(x) = mx + b \).

Algebraic manipulation involves rearranging expressions and equations to isolate variables or combine like terms. This process often includes a sequence of operations like addition, subtraction, multiplication, and division to simplify the expressions.
In the exercise, we use several algebraic manipulation techniques to solve the inequality.
First, we set up the inequality:\[ 8x - 9 \text{ ≤ } 3x - 11 \]Next, we move the \( x \)-terms to one side by subtracting \( 3x \) from both sides:
\[ 8x - 3x - 9 \text{ ≤ } -11 \]This simplifies to: \[ 5x - 9 \text{ ≤ } -11 \]Then, we isolate the constant term by adding 9 to both sides:
\[ 5x - 9 + 9 \text{ ≤ } -11 + 9 \]Which simplifies to:
\[ 5x \text{ ≤ } -2 \]Finally, we solve for \( x \) by dividing both sides by 5: \[ x \text{ ≤ } -\frac{2}{5} \]

Solving inequalities involves finding the values of the variable that make the inequality true. Inequalities indicate that one side is less than, greater than, or equal to the other side. For the given problem, the statement \( f(x) ≤ g(x) \) translates to:
\[ 8x - 9 ≤ 3x - 11 \]We solve this by isolating \( x \). Make sure to perform the same operation on both sides to maintain the inequality’s balance.
During our solution, each step included operations that help us break down the inequality further until \( x \) is alone. Here’s a breakdown of those key steps:

• Moved all \( x \)-terms to one side
• Isolated the constant term on the other side
• Divided to solve for \( x \)

The final solution \( x ≤ -\frac{2}{5} \) means all values of \( x \) less than or equal to -\( \frac{2}{5} \) make the inequality true.