$$ 1 \leftarrow $$ Solve each of the following inequalities for $$ x $$, where $$ x $$ is an integer. $$ \begin{array}{ll} ext { a. } 6^{x} \geq 216 & ext { c. } 5^{3 x} \leq 625 \ ext { b. } 4^{x} \geq 8 & ext { d. } 7^{x}

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$$ 1 \leftarrow $$ Solve each of the following inequalities for $$ x $$, where $$ x $$ is an integer. $$ \begin{array}{ll}	ext { a. } 6^{x} \geq 216 & 	ext { c. } 5^{3 x} \leq 625 \ 	ext { b. } 4^{x} \geq 8 & 	ext { d. } 7^{x}<1\end{array} $$ a. $$ \square $$ (Type an inequality.) b. $$ \square $$ (Type an inequality.) c. $$ \square $$ (Type an inequality.) d. $$ \square $$ (Type an inequality.)

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a. We note that $$216=6^3$$, so the inequality becomes $$ 6^x \geq 6^3. $$ Since the base $$6>1$$ is increasing, it follows that $$ x\geq 3. $$ b. Rewrite $$4^x$$ and $$8$$ in terms of base $$2$$: $$ 4^x=(2^2)^x=2^{2x} \quad \text{and} \quad 8=2^3. $$ The inequality becomes $$ 2^{2x}\geq 2^3. $$ Since the base $$2>1$$ is increasing, we get $$ 2x\geq 3 \quad \Longrightarrow \quad x\geq \frac{3}{2}. $$ As $$x$$ is an integer, $$x\geq 2$$. c. Express $$625$$ as a power of $$5$$: $$ 625=5^4. $$ The inequality becomes $$ 5^{3x}\leq 5^4. $$ Since $$5>1$$, this implies $$ 3x\leq 4 \quad \Longrightarrow \quad x\leq \frac{4}{3}. $$ As $$x$$ is an integer, $$x\leq 1$$. d. Since $$7>1$$, for positive exponents $$7^x>1$$. We have $$ 7^x < 1, $$ which holds only when $$ x<0. $$ a. $$x \geq 3$$ b. $$x \geq 2$$ c. $$x \leq 1$$ d. $$x < 0$$ a. $$x \geq 3$$ b. $$x \geq 2$$ c. $$x \leq 1$$ d. $$x < 0$$

Solve each of the following inequalities for , where is an integer.

a. (Type an inequality.)
b. (Type an inequality.)
c. (Type an inequality.)
d. (Type an inequality.)

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Solve each of the following inequalities for , where is an integer. a. (Type an inequality.) b. (Type an inequality.) c. (Type an inequality.) d. (Type an inequality.)