Certainly! Let's solve the inequality [tex]\(x^2 - 21 < 4x\)[/tex] step-by-step.
First, we want to rearrange the inequality to get everything on one side:
[tex]\[x^2 - 21 < 4x\][/tex]
Subtract [tex]\(4x\)[/tex] from both sides to set the inequality to zero:
[tex]\[x^2 - 4x - 21 < 0\][/tex]
Now, we factor the quadratic expression [tex]\(x^2 - 4x - 21\)[/tex]. We look for two numbers that multiply to [tex]\(-21\)[/tex] and add to [tex]\(-4\)[/tex].
Those numbers are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex], because:
[tex]\[ -7 \times 3 = -21 \][/tex]
[tex]\[ -7 + 3 = -4 \][/tex]
So the quadratic expression factors as follows:
[tex]\[(x - 7)(x + 3)\][/tex]
Thus, the inequality [tex]\(x^2 - 4x - 21 < 0\)[/tex] can be written in its factored form:
[tex]\[(x - 7)(x + 3) < 0\][/tex]
Next, we find the critical points where the expression equals zero:
[tex]\[(x - 7) = 0\][/tex]
[tex]\[(x + 3) = 0\][/tex]
This gives us the critical points [tex]\(x = 7\)[/tex] and [tex]\(x = -3\)[/tex]. These points divide the number line into three intervals:
1. [tex]\(x < -3\)[/tex]
2. [tex]\(-3 < x < 7\)[/tex]
3. [tex]\(x > 7\)[/tex]
We need to test a value from each interval to see where the inequality holds true:
1. For the interval [tex]\(x < -3\)[/tex], let's choose [tex]\(x = -4\)[/tex]:
[tex]\[(x - 7)(x + 3) = (-4 - 7)(-4 + 3) = (-11)(-1) = 11\][/tex]
Since [tex]\(11 > 0\)[/tex], the inequality does not hold in this interval.
2. For the interval [tex]\(-3 < x < 7\)[/tex], let's choose [tex]\(x = 0\)[/tex]:
[tex]\[(x - 7)(x + 3) = (0 - 7)(0 + 3) = (-7)(3) = -21\][/tex]
Since [tex]\(-21 < 0\)[/tex], the inequality holds in this interval.
3. For the interval [tex]\(x > 7\)[/tex], let's choose [tex]\(x = 8\)[/tex]:
[tex]\[(x - 7)(x + 3) = (8 - 7)(8 + 3) = (1)(11) = 11\][/tex]
Since [tex]\(11 > 0\)[/tex], the inequality does not hold in this interval.
Therefore, the solution to the inequality [tex]\(x^2 - 21 < 4x\)[/tex] is:
[tex]\[-3 < x < 7\][/tex]
In interval notation, the solution is:
[tex]\(\boxed{(-3, 7)}\)[/tex]
