91Ó°ÊÓ

Graphing functions, particularly absolute value functions, creates a clear visual of an equation. Understanding how to graph the function \(y = |x + 3|\) gives us insight into solution behaviors. The absolute value function \(y = |x + 3|\) resembles a "V" shape graph that is symmetric about the vertical line through its vertex.
The vertex of the graph is the lowest point since \(y\) can never be negative. For \(y = |x + 3|\), the vertex is at \(x = -3\). From this point, the line goes upwards to the right and upwards to the left.
To solve the inequality \(|x+3| \geq 5\), we need to find where the graph lies outside the line \y = 5\.By observing where the V-shape function intersects or lies above the line, the solution becomes more intuitive.

Interval notation is a concise way of expressing subsets of the real line. For the inequality \( |x+3| \geq 5 \), once solutions are identified, we can express these solutions succinctly.
Interval notation utilizes brackets:

• Square brackets \[\text{[} ]\] indicate that an endpoint is included in the set, reflecting a 'greater than or equal to' or 'less than or equal to' situation.
• Parentheses \(\text{(} )\) signal that an endpoint is not included, showing a 'greater than' or 'less than' situation.

In our example, the solution is expressed as \( (-\infty, -8] \cup [2, \infty) \).This notation indicates two solution regions, one reaching up to and including -8, and another beginning at and including 2 and extending to infinity.

Solving absolute value inequalities involves simplifying expressions to find the range of values that satisfy them. The expression \( |x + 3| \geq 5 \) means that \( x + 3 \) can either be more than 5 or less than -5.
To tackle this inequality, we split it into two separate inequalities:

• \ x + 3 \geq 5 \ which simplifies to \( x \geq 2 \)
• \ x + 3 \leq -5 \ which simplifies to \( x \leq -8 \)

Each inequality is solved by isolating \( x \) on one side. These two distinct regions where the inequality holds form the solutions we seek.
Using inequalities, we carve out specific intervals on the number line that satisfy the original inequality conditions.

A number line offers a visual method to represent solutions of inequalities. For the inequality \( |x+3| \geq 5 \), once solutions are derived, placing them on a number line helps validate and clarify the solution intervals.
For our exercise, consider the intervals:

• The interval \( x \geq 2 \) is represented by a line extending rightwards starting from 2, including 2, shown with a filled dot.
• The interval \( x \leq -8 \) is depicted by a line extending leftwards from -8, including -8 fully shaded.

On the number line, these intervals appear disjoint, mirroring the unique regions where the inequality conditions are satisfied.By visually showing these areas, it becomes easier to connect the number line representation back to the frame of interval notation.