6) (1(2)/(7))^x^(2-4)leqslant 1

To solve the inequality \((1\frac{2}{7})^{x^2 - 4} \leq 1\), we first need to express the mixed number \(1\frac{2}{7}\) as an improper fraction.<br /><br />\[ 1\frac{2}{7} = \frac{7 \cdot 1 + 2}{7} = \frac{9}{7} \]<br /><br />So the inequality becomes:<br /><br />\[ \left(\frac{9}{7}\right)^{x^2 - 4} \leq 1 \]<br /><br />Next, we analyze the expression \(\left(\frac{9}{7}\right)^{x^2 - 4}\). For this expression to be less than or equal to 1, the exponent \(x^2 - 4\) must be less than or equal to 0 because \(\frac{9}{7} > 1\).<br /><br />\[ x^2 - 4 \leq 0 \]<br /><br />Solving this quadratic inequality:<br /><br />\[ x^2 \leq 4 \]<br /><br />This gives us:<br /><br />\[ -2 \leq x \leq 2 \]<br /><br />Therefore, the solution to the inequality \((1\frac{2}{7})^{x^2 - 4} \leq 1\) is:<br /><br />\[ -2 \leq x \leq 2 \]