Problem 24 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ -112

Step-by-Step Solution

Answer

The interval is \((-112, -12)\).

Step 1: Identify the Inequality

The inequality given is \(-112 < x < -12\). This is a compound inequality, indicating that \(x\) is greater than \(-112\) and less than \(-12\).

Step 2: Plot Points on the Number Line

Draw a number line. Mark the points \(-112\) and \(-12\) on this number line. These are just markers to define the boundary of our solution.

Step 3: Indicate Open Intervals

Since the inequality is strict (less than \(-112\) and \(-12\) with no equal sign), use open circles (or hollow dots) at both \(-112\) and \(-12\) on the number line.

Step 4: Shade the Solution Region

Shade the region between \(-112\) and \(-12\) on the number line, indicating that \(x\) can be any value between these two points, but not including the points themselves.

Step 5: Write Interval Notation

The interval notation for the solution is \((-112, -12)\), meaning all numbers between \(-112\) and \(-12\), excluding \(-112\) and \(-12\) themselves.

Key Concepts

Number Line

A number line is a visual representation of numbers laid out in a straight, horizontal line. It helps us understand how numbers relate to each other by positioning each number at equal distances from one another.
- When dealing with inequalities, a number line can effectively illustrate which numbers satisfy the inequality conditions.
- On a number line, smaller numbers appear to the left of bigger numbers.
- To graph an inequality like \(-112 < x < -12\), we mark the boundary points, \(-112\) and \(-12\), to identify the limits of our solution. In summary, a number line serves as a straightforward way to visualize solutions to inequalities.

Compound Inequality

A compound inequality involves two separate inequalities joined by the word 'and' or 'or.' Here, the inequality \(-112 < x < -12\) is a 'compound' because it specifies two conditions at once: - \(x\) must be greater than \(-112\).
- \(x\) must also be less than \(-12\).
The solution to a compound inequality is the set of values that satisfy both conditions simultaneously. Using a compound inequality helps to define a range of possible values—and understanding this range is key to graphing and solving inequalities effectively.

Interval Notation

Interval notation is a succinct way to express a set of numbers that fall within a certain range.For the inequality \(-112 < x < -12\), the interval notation is \((-112, -12)\). Here's how it works:- The parentheses indicate that the boundary numbers \(-112\) and \(-12\) are not included in the set. - Unlike with brackets \([]\), which would mean inclusion, parentheses always signify exclusion.
Interval notation allows for a clean and simplified way to convey the results of an inequality.

Open Intervals

Open intervals focus on a range between two numbers, excluding the endpoints themselves. In graphing, this is shown by:- Using open circles (or hollow dots) at \(-112\) and \(-12\) on the number line. - The open circles show that these points are boundaries but not part of the solution set.An open interval like \((-112, -12)\) means any number between \(-112\) and \(-12\) is included, but not \(-112\) or \(-12\) themselves.
Understanding open intervals is crucial when interpreting inequalities and accurately graphing them on a number line.