To solve the equation [tex]\((2x - 3)(x + 4) = 0\)[/tex], we need to find the roots of the equation by setting each factor equal to zero.
1. Set [tex]\(2x - 3 = 0\)[/tex]:
[tex]\[
2x - 3 = 0
\][/tex]
Add 3 to both sides:
[tex]\[
2x = 3
\][/tex]
Now, divide both sides by 2:
[tex]\[
x = \frac{3}{2}
\][/tex]
2. Set [tex]\(x + 4 = 0\)[/tex]:
[tex]\[
x + 4 = 0
\][/tex]
Subtract 4 from both sides:
[tex]\[
x = -4
\][/tex]
Thus, the roots of the equation are [tex]\(x = \frac{3}{2}\)[/tex] and [tex]\(x = -4\)[/tex]. Let's denote [tex]\(a = \frac{3}{2}\)[/tex] and [tex]\(b = -4\)[/tex].
We are asked to find the value of [tex]\(-a - b\)[/tex].
Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the expression:
[tex]\[
-a - b = -\left(\frac{3}{2}\right) - (-4)
\][/tex]
First, compute the negation of [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
-\left(\frac{3}{2}\right) = -\frac{3}{2}
\][/tex]
Next, simplify the negation of [tex]\(-4\)[/tex]:
[tex]\[
-(-4) = 4
\][/tex]
Now, combine these results:
[tex]\[
-a - b = -\frac{3}{2} + 4
\][/tex]
Convert 4 to a fraction with a common denominator:
[tex]\[
4 = \frac{8}{2}
\][/tex]
Add the fractions:
[tex]\[
-\frac{3}{2} + \frac{8}{2} = \frac{-3 + 8}{2} = \frac{5}{2}
\][/tex]
Therefore, the value of [tex]\(-a - b\)[/tex] is:
[tex]\[
\boxed{\frac{5}{2}}
\][/tex]
