91Ó°ÊÓ

Absolute value is an important concept in mathematics that represents the distance of a number from zero on a number line. In our exercise, the expression \(|2x + 7| \leq 13\) involves an absolute value inequality. The key idea here is that the absolute value of any number or algebraic expression is always non-negative, meaning it is zero or positive.

So, when you see an inequality with an absolute value, like \(|A| \leq B\), you should interpret it as the distance between the expression inside the absolute value and zero being less than or equal to \(B\). The absolute value inequality \(|A| \leq B\) can be split into two separate inequalities without absolute values: \(A \leq B\) and \(A \geq -B\). In our exercise, this helps us convert \(|2x + 7| \leq 13\) into two individual inequalities, allowing us to solve for \(x\) by standard algebraic methods.

Interval notation is a concise way of describing sets of numbers, particularly solutions to inequalities. For the problem at hand, after solving the inequality \(|2x + 7| \leq 13\), we find that the values of \(x\) are between -10 and 3.

• An interval that contains both endpoints, like -10 and 3 in this case, is called a "closed interval."
• Closed intervals are represented by square brackets, which indicate that the endpoints are included in the solution.
• In interval notation, -10 and 3 become \([-10, 3]\).

This notation is advantageous because it provides a clear summary of all the values in the solution set, without having to use a more verbose form such as inequalities. It's like a shortcut that tells us we're considering every value between -10 and 3, including these endpoints.

Graphing inequalities on a number line is a visual method to express the solutions of the inequality. It involves shading a section of the number line to represent all possible values of the variable that satisfy the inequality.

For the inequality solution found in our exercise, \(-10 \leq x \leq 3\), graphing involves a few simple steps:

• Locate and mark the endpoints -10 and 3 on the number line.
• Use solid dots at -10 and 3 because the inequality includes these values (indicated by the \(\leq\) sign).
• Shade the entire region between the points, indicating every number in that range is a solution to the original inequality.

This visual representation makes it easy to understand which values are included, providing a clear connection between the written solution in interval notation and the raw numbers on the number line.