Find inverse Laplace transform ${L}^{-1}\left\{\frac{s-5}{{s}^{2}+5s-24}\right\}$ Please provide supporting details for your answer
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Step 1
The following formulae are used in solving the given problem.
${L}^{-1}f\left(x\right)+g\left(x\right)={L}^{-1}\left(f\left(x\right)\right)+{L}^{-1}\left(g\left(x\right)\right)$
${L}^{-1}\left(a\cdot f\left(x\right)\right)=a{L}^{-1}\left(f\left(x\right)\right)$
${L}^{-1}\left(\frac{1}{s+a}\right)={e}^{-at}$
Step 2
Evaluate the inverse Laplace transform of the given function as follows.

${L}^{-1}\left\{-\frac{2}{11\left(s-3\right)}\right\}+{L}^{-1}\left\{\frac{13}{11\left(s+8\right)}\right\}$
$-\frac{2}{11}{L}^{-1}\left\{1\left(s-3\right)\right\}+\frac{13}{11}{L}^{-1}\left\{\frac{1}{s+8}\right\}$
$-\frac{2}{11}{e}^{3t}+\frac{13}{11}{e}^{-8t}$
Therefore, the inverse Laplace transform of the given function is $-\frac{2}{11}{e}^{3t}+\frac{13}{11}{e}^{-8t}$