Determine the values of k for which the equation has no solutions: vert 2x+7vert -kleqslant 5 klt -5 kgt 5 klt 5 kgt -5

To determine the values of k for which the equation has no solutions, we need to analyze the given inequality:<br /><br />$\vert 2x+7\vert -k\leqslant 5$<br /><br />First, let's isolate the absolute value expression:<br /><br />$\vert 2x+7\vert \leqslant 5+k$<br /><br />Now, we need to consider the two cases for the absolute value expression:<br /><br />Case 1: $2x+7\leqslant 5+k$<br />$2x\leqslant -2-k$<br />$x\leqslant -1-\frac{k}{2}$<br /><br />Case 2: $-(2x+7)\leqslant 5+k$<br />$-2x-7\leqslant 5+k$<br />$-2x\leqslant 12+k$<br />$x\geqslant -6-\frac{k}{2}$<br /><br />For the equation to have no solutions, there should be no values of x that satisfy both cases. This means that the range of x for both cases should not overlap.<br /><br />In Case 1, the range of x is $x\leqslant -1-\frac{k}{2}$, and in Case 2, the range of x is $x\geqslant -6-\frac{k}{2}$.<br /><br />For these ranges to not overlap, the value of k must be such that $-1-\frac{k}{2} < -6-\frac{k}{2}$. Solving this inequality gives $k > 5$.<br /><br />Therefore, the correct answer is: $k\gt 5$.