91Ó°ÊÓ

Interval notation is a simple and convenient way to express a range of values, typically numbers along the real number line, that satisfy a particular inequality. It uses brackets and parentheses to show whether endpoints are included in the set.

• Square Brackets [ ]: Indicate the endpoint is included in the interval, known as a closed interval.
• Parentheses ( ): Indicate the endpoint is not included, known as an open interval.

Let's take our inequality solution from Part a, \(3 \leq x \leq 7\), as an example. It is expressed in interval notation as \[3, 7\], where both 3 and 7 are included in the solution set because they satisfy the inequality.

For the inequality solution in Part b, \(x < 3\) or \(x > 7\), we use \((-\infty, 3) \cup (7, \infty)\). Here, the use of parentheses shows the numbers 3 and 7 are not part of the solution. Infinity is always represented with parentheses because it is a concept, not a specific number, and thus can't be included.

Graphing inequalities on a number line provides a visual representation that makes it easier to understand which values satisfy the inequality. Here's how you can achieve this step by step:

• Closed Circle: On the number line, use a closed (filled) circle to show that a number is part of the solution. This equals the square bracket in interval notation.
• Open Circle: Use an open circle for numbers not included in the solution, which corresponds with the round parenthesis in interval notation.

For the absolute value inequality \[3, 7\], place closed circles at both 3 and 7, and shade the line segment connecting these points. This shading means all numbers between and including 3 and 7 are solutions.

In Part b, \((-\infty, 3) \cup (7, \infty)\), plot open circles at 3 and 7, and shade the line segments extending infinitely to the left of 3 and right of 7. This illustrates all numbers except those in the interval \[3, 7\] are solutions.

Solving absolute value inequalities involves understanding the property that absolute values measure "distance" from zero, which is always a non-negative number.

When we have an inequality like \|8x - 40| \leq 16\, we need to consider what it means:

• We break down the inequality into two parts: \(8x - 40 \leq 16\) and \(8x - 40 \geq -16\)
• Solving each gives us a range, resulting in \(3 \leq x \leq 7\)

The intuitive way to interpret absolute inequality \|8x - 40| \geq 16\ is that the values are outside the "window" of the central range specified in Part a. The solution entails the complements of that range, leading to \(x < 3\) or \(x > 7\). This expresses two conditions combined as \((-\infty, 3) \cup (7, \infty)\). Each part of the inequality represents a segment outside our initial solution set's bounds.