To solve the inequality [tex]\(4x^2 - 25 < 0\)[/tex], follow these detailed steps:
1. Rewrite the inequality:
Start with the given inequality:
[tex]\[
4x^2 - 25 < 0
\][/tex]
2. Rewrite it as a quadratic equation:
Consider the corresponding equation:
[tex]\[
4x^2 - 25 = 0
\][/tex]
3. Factorize the quadratic equation:
You can factorize [tex]\(4x^2 - 25\)[/tex] as a difference of squares:
[tex]\[
4x^2 - 25 = (2x - 5)(2x + 5)
\][/tex]
So the equation becomes:
[tex]\[
(2x - 5)(2x + 5) = 0
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
2x - 5 = 0 \quad \text{or} \quad 2x + 5 = 0
\][/tex]
Solving these gives:
[tex]\[
2x = 5 \quad \text{or} \quad 2x = -5
\][/tex]
So,
[tex]\[
x = \frac{5}{2} \quad \text{or} \quad x = \frac{-5}{2}
\][/tex]
Therefore, [tex]\(x = 2.5\)[/tex] or [tex]\(x = -2.5\)[/tex].
5. Determine intervals to check the signs:
Based on the roots [tex]\(x = 2.5\)[/tex] and [tex]\(x = -2.5\)[/tex], divide the number line into intervals:
[tex]\[
(-\infty, -2.5), \quad (-2.5, 2.5), \quad (2.5, \infty)
\][/tex]
6. Test points in the intervals to see where the inequality is satisfied:
- For the interval [tex]\((-∞, -2.5)\)[/tex]:
Choose a test point, say [tex]\(x = -3\)[/tex]:
[tex]\[
4(-3)^2 - 25 = 4(9) - 25 = 36 - 25 = 11
\][/tex]
Since 11 is greater than 0, this interval does not satisfy the inequality.
- For the interval [tex]\((-2.5, 2.5)\)[/tex]:
Choose a test point, say [tex]\(x = 0\)[/tex]:
[tex]\[
4(0)^2 - 25 = 0 - 25 = -25
\][/tex]
Since -25 is less than 0, this interval satisfies the inequality.
- For the interval [tex]\((2.5, ∞)\)[/tex]:
Choose a test point, say [tex]\(x = 3\)[/tex]:
[tex]\[
4(3)^2 - 25 = 4(9) - 25 = 36 - 25 = 11
\][/tex]
Since 11 is greater than 0, this interval does not satisfy the inequality.
7. Determine which intervals satisfy the inequality:
From the interval test, the single interval that satisfies the inequality [tex]\(4x^2 - 25 < 0\)[/tex] is:
[tex]\[
(-2.5, 2.5)
\][/tex]
So, the solution to the inequality [tex]\(4x^2 - 25 < 0\)[/tex] is:
[tex]\[
-2.5 < x < 2.5
\][/tex]
