Express the interval in terms of an inequality involving...

Key Concepts

Absolute Value Inequality

An absolute value inequality uses the absolute value function to define a range of values based on their distance from a particular number. Such inequalities are often rewritten in the form |x - c| < r, where c is the center of the interval and r is the radius, which provides a clear geometric interpretation on the number line as all numbers within a distance r from c.

Interval Notation

Interval notation is a method of writing subsets of the real numbers as intervals between two endpoints. It provides a concise way to describe a range of values, using parentheses to denote that an endpoint is not included and brackets to indicate inclusion.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of direction. It is a measure of magnitude and is always non-negative, symbolized by vertical bars around a number or expression.

Inequalities

Inequalities are mathematical statements that express the relative size or order of two values. They articulate when one quantity is less than, greater than, or equal to another, and are foundational in describing ranges and intervals of numbers.

Transcript

00:01 Hi, today we will be solving the following problem.

00:04 Express the interval in terms of an inequality involving absolute value.

00:09 And the interval that we are given is 0 to 4.

00:12 So first let's try defining some of our key terms.

00:16 Firstly, interval.

00:17 An interval is just a set of numbers.

00:21 For example, the interval 0 to 4 is a set of all numbers in between 0 and 4.

00:26 This includes numbers such as 1, 2, 3, even 3 .999.

00:32 However, it is worth noting that 0 and 4 are not included in this interval.

00:38 Only numbers that are in between those values are included.

00:43 Now let's define an inequality.

00:46 An inequality is a restriction on a set of numbers.

00:51 For example, if i were to say x is less than 2, x being any arbitrary value from negative infinity to positive infinity, that would be to say any value of x such as 1 and negative 10 would be valid for this inequality as long as it's real and less than 2.

01:19 So numbers such as 3 and 102 billion would not be included in this inequality.

01:26 Now let's define absolute value.

01:30 Absolute value is the distance a number is from the number zero.

01:37 So for example, if i were to try to find the absolute value of a number such as negative 2, i would look on the number line and i would look for the distance negative 2 is from 0.

01:55 Now negative 2 is 1 -2 spaces from 0.

02:02 So that means that the absolute value of negative 2.

02:05 And i can express the absolute value of a number by putting two vertical lines on both sides of it.

02:12 The absolute value of negative 2 is just 2.

02:17 It's pretty simple.

02:21 So now let's get to solving this problem.

02:25 The first step would be to write an inequality.

02:30 That is representative of this interval.

02:39 Now we know that we want all numbers that are between 0 and 4.

02:46 So we would do this by expressing that we want all values, and x would be representing any value.

02:55 It's a variable that represents any value, that is less than 4 and greater than 0.

03:07 This is an inequality that can express, all of the numbers that we want to deal with.

03:13 Any number that is less than 4 and greater than 0 is included in our interval.

03:19 Now the next step is to find the midpoint of our interval.

03:30 We can do this again by visualizing our interval on a number line.

03:36 So if we have the number line over here, we would want all numbers between 0 and 4.

03:47 The midpoint would be the number that is in the dead center of this interval.

03:53 We can solve for the midpoint by taking the average of our two bounds, which are 4 and 0.

04:01 The bounds are just the ends of the interval.

04:03 They're what bound the sides, the set of numbers that we want to define.

04:10 So our midpoint would be the average of 4 and 0.

04:15 So 4 plus 0 over 2.

04:20 And that's equal to 4 over 2.

04:26 And that is finally equal to 2.

04:30 So the midpoint of our interval is 2...

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