<p>In mathematics, the integral inequalities are crucial. Research on this topic has recently been conducted using the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{q}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> </math></EquationSource> </InlineEquation>-, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\varvec{p},\varvec{q})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-, and symmetric <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varvec{q}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> </math></EquationSource> </InlineEquation>-calculus. In this manuscript, first, we consider the left endpoint of the interval and introduce the new concept of integration and derivatives in dual basic symmetric quantum calculus. Moreover, we derive symmetric post-quantum variants of integral inequalities, i.e., <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\widetilde{(\varvec{p},\varvec{q})}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>-H<InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\ddot{o}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>o</mi> <mo>¨</mo> </mover> </math></EquationSource> </InlineEquation>lder, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\widetilde{(\varvec{p},\varvec{q})}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>- Minkowski, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\widetilde{(\varvec{p},\varvec{q})}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>-Hermite–Hadamard, and some kinds of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\widetilde{(\varvec{p},\varvec{q})}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>-Hermite–Hadamard inequalities. Finally, we give a graphical analysis of our obtained results by selecting certain parameters in different variants of newly introduced <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\widetilde{(\varvec{p},\varvec{q})}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">q</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>-Hermite–Hadamard-type inequalities via support line.</p>