\( 1 \leftarrow \quad \) Solve each of the following inequalities for \( x \), where \( x \) is an integer. \( \begin{array}{ll}\text { a. } 2^{x} \leq 64 & \text { c. } 3^{4 x}>243 \\ \text { b. } 8^{x}<4 & \text { d. } 2^{x}>1\end{array} \) a. \( x \leq 6 \) (Type an inequality. Use integers for any numbers in the expression.) b. \( x \leq 0 \) (Type an inequality. Use integers for any numbers in the expression.) c. \( \square \) (Type an inequality. Use integers for any numbers in the expression.)

Let's solve each inequality step by step. ### a. Solve \( 2^{x} \leq 64 \) 1. Rewrite \( 64 \) as a power of \( 2 \): \[ 64 = 2^6 \] 2. Set up the inequality: \[ 2^{x} \leq 2^6 \] 3. Since the bases are the same, we can compare the exponents: \[ x \leq 6 \] ### b. Solve \( 8^{x} < 4 \) 1. Rewrite \( 8 \) and \( 4 \) as powers of \( 2 \): \[ 8 = 2^3 \quad \text{and} \quad 4 = 2^2 \] 2. Substitute into the inequality: \[ (2^3)^{x} < 2^2 \] 3. Simplify the left side: \[ 2^{3x} < 2^2 \] 4. Compare the exponents: \[ 3x < 2 \] 5. Solve for \( x \): \[ x < \frac{2}{3} \] Since \( x \) must be an integer, the largest integer satisfying this inequality is: \[ x \leq 0 \] ### c. Solve \( 3^{4x} > 243 \) 1. Rewrite \( 243 \) as a power of \( 3 \): \[ 243 = 3^5 \] 2. Set up the inequality: \[ 3^{4x} > 3^5 \] 3. Since the bases are the same, we can compare the exponents: \[ 4x > 5 \] 4. Solve for \( x \): \[ x > \frac{5}{4} \] Since \( x \) must be an integer, the smallest integer satisfying this inequality is: \[ x \geq 2 \] ### d. Solve \( 2^{x} > 1 \) 1. Rewrite \( 1 \) as a power of \( 2 \): \[ 1 = 2^0 \] 2. Set up the inequality: \[ 2^{x} > 2^0 \] 3. Compare the exponents: \[ x > 0 \] ### Summary of Solutions - a. \( x \leq 6 \) - b. \( x \leq 0 \) - c. \( x \geq 2 \) - d. \( x > 0 \) Thus, the final answers are: - a. \( x \leq 6 \) - b. \( x \leq 0 \) - c. \( x \geq 2 \)