1) Calculate the following: &i. Calculate $1^2 - 2^3 \times 3 + 8 \div 4 - (3^3 + 2)$: $1^2 = 1$ $2^3 = 8$ $8 \times 3 = 24$ $8 \div 4 = 2$ $3^3 = 27$ Sum inside parentheses: $27 + 2 = 29$ Expression becomes: $1 - 24 + 2 - 29$ Simplify: $1 - 24 = -23$ Then, $-23 + 2 = -21$ Finally, $-21 - 29 = -50$ Answer: $-50$ &ii. Calculate $10 - 10 \times 10 + 10 - (10 + 10 \times 10)$: First compute $10 \times 10 = 100$ Inside parentheses: $10 + 100 = 110$ Expression: $10 - 100 + 10 - 110$ Simplify: $10 - 100 = -90$ Then, $-90 + 10 = -80$ Then, $-80 - 110 = -190$ Answer: $-190$ &iii. Calculate $(3 - 5)^3 - 3 + 3^2 \times (5 + 1)$: $3 - 5 = -2$ $(-2)^3 = -8$ $3^2 = 9$ $5 + 1 = 6$ $9 \times 6 = 54$ Expression: $-8 - 3 + 54$ Simplify: $-8 - 3 = -11$ Then, $-11 + 54 = 43$ Answer: $43$ &iv. Calculate $2 - (2 + 2 \times 2^2)^2 + 2 \div 2 + 2^2$ Calculate powers and multiplication inside parentheses: $2^2 = 4$ $2 \times 4 = 8$ Inside parentheses: $2 + 8 = 10$ Square: $10^2 = 100$ $2 \div 2 = 1$ $2^2 = 4$ Expression: $2 - 100 + 1 + 4$ Simplify: $2 - 100 = -98$ Then, $-98 + 1 = -97$ Then, $-97 + 4 = -93$ Answer: $-93$ &v. Calculate $\left[11 - 4(2 - 33)\right] \div 37$ Calculate inside parentheses: $2 - 33 = -31$ Multiply: $4 \times (-31) = -124$ Inside brackets: $11 - (-124) = 11 + 124 = 135$ Divide by 37: $135 \div 37 = \frac{135}{37}$ (approx. 3.65) Answer: $\frac{135}{37}$ or approximately $3.65$ &vi. Calculate $\left[7 + 3(23 - 1)\right] \div 21$ Calculate inside parentheses: $23 - 1 = 22$ Multiply: $3 \times 22 = 66$ Inside brackets: $7 + 66 = 73$ Divide by 21: $73 \div 21 = \frac{73}{21}$ (approx. 3.48) Answer: $\frac{73}{21}$ or approximately $3.48$ 2) Solve the following equations: i. Solve $2(3x - 4) = 4x + 10$ Expand left side: $6x - 8 = 4x + 10$ Subtract $4x$ from both sides: $6x - 4x - 8 = 10$ Simplify: $2x - 8 = 10$ Add 8 to both sides: $2x = 18$ Divide both sides by 2: $x = 9$ Answer: $x = 9$ ii. Solve $|3x + 5| = 17$ Two cases: Case 1: $3x + 5 = 17$ Subtract 5: $3x = 12$ Divide by 3: $x = 4$ Case 2: $3x + 5 = -17$ Subtract 5: $3x = -22$ Divide by 3: $x = -\frac{22}{3}$ Answer: $x = 4$ or $x = -\frac{22}{3}$ iii. Solve $|x + 5| = -8$ Absolute value cannot be negative No solution Answer: No solution iv. Solve $|2x + 5| = x - 1$ Right side must be nonnegative: $x - 1 \geq 0 \Rightarrow x \geq 1$ Case 1: $2x + 5 = x - 1$ Subtract $x$: $x + 5 = -1$ Subtract 5: $x = -6$ Not valid since $x \geq 1$, discard Case 2: $-(2x + 5) = x - 1$ $-2x - 5 = x - 1$ Add $2x$: $-5 = 3x - 1$ Add 1: $-4 = 3x$ Divide by 3: $x = -\frac{4}{3}$ Not valid since $x \geq 1$, discard Answer: No solution v. Solve $|3x - 2| = 0$ Absolute value equals zero means inside is zero $3x - 2 = 0$ $3x = 2$ $x = \frac{2}{3}$ Answer: $x = \frac{2}{3}$ 3) Solve each inequality: 1. $6 < x + 3 < 8$ Subtract 3: $3 < x < 5$ Answer: $x \in (3,5)$ 2. $7 < x + 5 < 11$ Subtract 5: $2 < x < 6$ Answer: $x \in (2,6)$ 3. $|2x - 6| > 8$ Two cases: $2x - 6 > 8 \Rightarrow 2x > 14 \Rightarrow x > 7$ Or $2x - 6 < -8 \Rightarrow 2x < -2 \Rightarrow x < -1$ Answer: $x < -1$ or $x > 7$ 4. $|3x + 5| < 17$ Inside absolute value between $-17$ and $17$ $-17 < 3x + 5 < 17$ Subtract 5: $-22 < 3x < 12$ Divide 3: $-\frac{22}{3} < x < 4$ Answer: $x \in \left(-\frac{22}{3}, 4\right)$ 5. $|2(x - 1) + 4| \leq 8$ Inside absolute value: $2x - 2 + 4 = 2x + 2$ Inequality: $|2x + 2| \leq 8$ So, $-8 \leq 2x + 2 \leq 8$ Subtract 2: $-10 \leq 2x \leq 6$ Divide 2: $-5 \leq x \leq 3$ Answer: $x \in [-5, 3]$ 6. $|4x + 8| < 0$ Absolute value cannot be negative No solution Answer: No solution 7. $|x + 1| \geq 0$ Absolute value always nonnegative, so true for all $x$ Answer: $x \in \mathbb{R}$ (all real numbers) 8. $1 - (x + 3) \geq 4 - 2x$ Simplify left: $1 - x - 3 = -x - 2$ Inequality: $-x - 2 \geq 4 - 2x$ Add $2x$: $x - 2 \geq 4$ Add 2: $x \geq 6$ Answer: $x \geq 6$ 9. $5(3 - x) \leq 3x - 1$ Expand left: $15 - 5x \leq 3x - 1$ Add $5x$: $15 \leq 8x -1$ Add 1: $16 \leq 8x$ Divide by 8: $2 \leq x$ Answer: $x \geq 2$