Solve the inequlity x^2>16 and give the answer in interval notation.
Well, well, well, looks like we have an inequality to solve! Let's get crackin'!
To get started, we'll find the values of x that make x^2 greater than 16.
First, we take the square root of both sides of the inequality: √(x^2) > √(16).
Now, don't forget that the square root can be positive or negative. So, we'll have two cases to consider.
Case 1: √(x^2) > 4.
This means that x can be any value greater than 4. In interval notation, this is (4, ∞).
Case 2: -(√(x^2)) > 4 (notice the negative sign).
To get rid of the negative sign, we multiply both sides by -1: √(x^2) < -4.
Ah, here comes the trickster part! You see, the square root of a number is always non-negative, so it can never be less than -4.
Therefore, Case 2 gives us an empty set, represented by the symbol Ø.
So, my friend, the solution to x^2 > 16 in interval notation is (4, ∞). Ta-da!
To solve the inequality x^2 > 16, we need to find the values of x that make this inequality true.
First, let's determine the values of x that make x^2 = 16. Taking the square root of both sides, we have:
x = ±√16
Simplifying further, we get:
x = ±4
So, the values of x that make x^2 = 16 are x = -4 and x = 4.
To determine the values of x that make x^2 > 16, we need to consider two cases:
1. When x > 4: In this case, all values of x greater than 4 will satisfy x^2 > 16.
2. When x < -4: In this case, all values of x less than -4 will also satisfy x^2 > 16.
Therefore, the solution in interval notation is (-∞, -4) ∪ (4, ∞).
To solve the inequality x^2 > 16, we need to find the values of x that make this inequality true. Here's how we can proceed:
Step 1: Start by factoring the left side of the inequality, which is x^2. However, since the expression is already in a quadratic form, we don't need to factor it further.
Step 2: Rewrite the inequality with the factored expression: x^2 - 16 > 0.
Step 3: To solve this inequality, we can use the method of factoring. Notice that this quadratic inequality is in the form of a difference of squares. We can factor it as follows: (x - 4)(x + 4) > 0.
Step 4: Now we have a product of two expressions that is greater than zero. To determine the solution, we need to consider the sign of each factor individually.
(i) When (x - 4) > 0 and (x + 4) > 0:
- If (x - 4) > 0, then x > 4.
- If (x + 4) > 0, then x > -4.
(ii) When (x - 4) < 0 and (x + 4) < 0:
- If (x - 4) < 0, then x < 4.
- If (x + 4) < 0, then x < -4.
Step 5: We can summarize the signs and intervals in a sign chart:
-4 4
(-)---0---(+)
Step 6: From the sign chart, we can see that the inequality (x-4)(x+4) > 0 is satisfied when x is less than -4 or greater than 4. In interval notation, the solution is (-∞, -4) ∪ (4, ∞). This represents all the values of x that make x^2 > 16 true.
x^2>16
<=>
±x>4
<=>
+x>4 or -x>4
<=>
x∈(4,∞) or x∈(-∞-4)
<=>
x∈ (-∞,-4)∪(4,∞)