Question

Multiple Choice Which of the following could be a test number for the interval \(-2

Short answer

Option (c): 4

Step-by-step solution

Understand the Interval

The interval (-2<x<5denotes all numbers that are greater than -2 but less than 5. It does not include -2 and 5 themselves.

Test Each Option

Evaluate each given option to see if it falls within the interval (-2<x<5.

Option (a): -3

-3 is less than -2, so it is outside of the interval (-2<x<5.Therefore, -3 is not a test number.

Option (b): -2

-2 is equal to the lower bound of the interval, and since the interval does not include -2 (as indicated by the strict inequality '<'), -2 is not within the interval. Therefore, -2 is not a test number.

Option (c): 4

4 is greater than -2 and less than 5, so it falls within the interval (-2<x<5.Therefore, 4 is a test number.

Option (d): 7

7 is greater than 5, so it is outside of the interval (-2<x<5.Therefore, 7 is not a test number.

Conclusion

After testing all the options, the number that lies within the interval (-2<x<5 is 4.

Key concepts

Interval Notation

Interval notation is a way to describe a range of numbers along a number line. It uses brackets \(( \, ) \)\text{ or } \([ \, ] \) to indicate which values are included and which are not:

• Parentheses (\text{}) indicate that the number is not included in the interval.
• Brackets (\text{}) indicate that the number is included.

For example, \((-2< x < 5)\)\text{ means }x\text{ is between -2 and 5, but does not include -2 and 5}. This is called an open interval because it doesn't include the endpoints. If we wrote it as \([-2 \, 5]\)\text{, it would be a closed interval that includes -2 and 5}.

Test Numbers

When determining if a value falls within a specific interval, we use what are called 'test numbers.' Test numbers are values we check to see if they lie within the given range. For example, in the interval \((-2< x < 5)\)\text{}, you would test numbers like -3, -2, 4, and 7 to see if they fit in the interval. Each test number must be compared against the bounds of the interval to determine if it falls inside or outside of it.

Strict Inequalities

Strict inequalities use the symbols > and < to show that a value is greater than or less than another value, but without including that value. For instance, \(-2 < x < 5\)\text{} means that }x\text{ must be greater than -2 and less than 5. It cannot be equal to -2 or 5 because those points are not part of the interval. If we used the symbols }\textcolor{“blue”}{weighted-bold;}geql\boxed{and}\ shown as\ [\text, -2 \text less than x \text ends greater equal\ od \ in filled-box\text bounds by a compared be would you\ regular-span\ the in \text>,

Number Line

A number line visually represents numbers in a straight line. When working with intervals, you can use a number line to easily see where values fall. Draw a number line and mark the interval you are considering. For the interval \( (-2 < x < 5) \)\text, place open circles on -2 and 5 to show that they are not included. Highlight or shade the area between -2 and 5 to show the range of valid numbers. This helps in visualizing which numbers lie inside and outside the interval.