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In this paper, we focus on a fractional differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo> </mo><mn>0</mn><mi>C</mi></msubsup><msup><mi>D</mi><mi>α</mi></msup><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>q</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> with boundary value conditions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mi>δ</mi><mi>u</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><msup><mi>u</mi><mo>′</mo></msup><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mi>γ</mi><mspace width="0.166667em"></mspace><msup><mi>u</mi><mo>′</mo></msup><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The paper begins by pointing out the inadequacies of the study conducted by Ma and Yangin establishing Lyapunov-type inequalities. It then discusses the properties of its Green’s function and investigates extremum problems related to several linear functions. Finally, thorough classification and analysis of various cases for parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> are conducted. As a result, a comprehensive solution corresponding to the Lyapunov-type inequality is obtained.