Express the interval in terms of an inequality involving absolute value. (0,4)

Question

Express the interval in terms of an inequality involving absolute value.  (0,4)

Szeteib · Accepted Answer

Step 1
To express the given interval in terms of an inequality involving absolute value.
Step 2
Given information:
The interval is (0, 4).
Step 3
Calculation:
Let (a, b) is an open interval and r is the radius, and the midpoint is c, then the interval can be represented as
${x \in R | x - c ∣ < r}$
where $c = \frac{a + b}{2}$ and $r = \frac{b - a}{2}$
And the given interval is (0, 4).
So, midpoint is
$c = \frac{a + b}{2}$
$c = \frac{0 + 4}{2}$
$c = \frac{4}{2}$
$c = 2$
And radius is
$r = \frac{b - a}{2}$
$r = \frac{4 - 0}{2}$
$r = \frac{4}{2}$
$r = 2$
Step 4
So, the inequality involving absolute value will be
${x \in R | x - c ∣ < r}$
${x \in R | x - 2 ∣ < 2}$ $[s u b s t i t u t e v a l u e s o f r a n d c]$
Thus, the inequality is ${x \in R | x - 2 ∣ < 2}$

nick1337 · Accepted Answer

Answer:0&amp;lt;|x|&amp;lt;4Explanation:Step 1: Identify the absolute value expression for the lower bound of the interval, which is 0. In this case, the absolute value of 0 is simply 0 itself.Step 2: Identify the absolute value expression for the upper bound of the interval, which is 4. The absolute value of 4 can be written as |4|, which is also equal to 4.Step 3: Write the inequality by combining the absolute value expressions. Since the interval (0, 4) does not include the endpoints 0 and 4, we use strict inequality symbols. The absolute value expression for the lower bound, 0, should be less than the absolute value expression for the upper bound, 4. Therefore, the inequality is:0&amp;lt;|x|&amp;lt;4This inequality states that any value of x that satisfies the condition of being greater than 0 and less than 4 (but not including 0 and 4) will be within the interval (0, 4).

Don Sumner · Accepted Answer

To express the interval (0,4) in terms of an inequality involving absolute value, we can use the following steps:
Step 1: Recall that the interval (0,4) represents all real numbers greater than 0 and less than 4, but does not include the endpoints 0 and 4.
Step 2: Let's start by writing the inequality for the lower bound of the interval, which is 0. Since we want numbers greater than 0, the inequality will be strict. We can write this as:

| x | > 0

Step 3: Next, let's write the inequality for the upper bound of the interval, which is 4. Since we want numbers less than 4, the inequality will also be strict. We can write this as:

| x | < 4

Step 4: Now, to combine the two inequalities and express the interval (0,4), we use the logical 'and'' operator (∧) between them. This indicates that both inequalities must be satisfied simultaneously. The combined inequality is:

| x | > 0 and | x | < 4

Step 5: Simplifying the combined inequality, we can remove the absolute value notation by considering two cases: when x is positive and when x is negative.
Case 1: x > 0
When x is positive, the absolute value of x is equal to x. Thus, we can rewrite the inequality as:

x > 0 and x < 4

Step 6: Case 2: x < 0
When x is negative, the absolute value of x is equal to -x. Therefore, we need to reverse the inequality signs when expressing the inequality. The inequality becomes:

- x > 0 and - x < 4

To simplify this, we multiply both sides of the inequalities by -1, which reverses the inequality signs:

x < 0 and x > - 4

Step 7: Combining the results from both cases, we can write the final expression for the interval (0,4) in terms of an inequality involving absolute value as:

(x > 0 and x < 4) or (x < 0 and x > - 4)

We can further simplify this expression as:

0 < x < 4 or - 4 < x < 0

Vasquez · Accepted Answer

The interval (0,4) can be expressed as an inequality involving absolute value as follows:|x|&amp;lt;4This inequality states that the absolute value of x is less than 4.

Express the interval in terms of an inequality involving absolute value. (0,4)

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