1. Stating the problems: We are solving for $x$ in the absolute value equations: a) $|x| = 3$ b) $|x+1| = 5$ c) $|x-2| = 3$ d) $|2x + 1| = 7$ e) $|x^{2026} + 2026^{100}| = -1$ 2. Solving equation a) $|x|=3$: Recall the absolute value property: $|A|=B$ implies $A = B$ or $A = -B$. Thus, $x = 3$ or $x = -3$. 3. Solving equation b) $|x+1| = 5$: i) $x + 1 = 5$, so $x = 5 - 1 = 4$. ii) $x + 1 = -5$, so $x = -5 - 1 = -6$. 4. Solving equation c) $|x-2| = 3$: i) $x - 2 = 3$, so $x = 3 + 2 = 5$. ii) $x - 2 = -3$, so $x = -3 + 2 = -1$. 5. Solving equation d) $|2x + 1| = 7$: i) $2x + 1 = 7$, so $2x = 6$, and thus $x = 3$. ii) $2x + 1 = -7$, so $2x = -8$, and thus $x = -4$. 6. Solving equation e) $|x^{2026} + 2026^{100}| = -1$: Since absolute value is always non-negative and never negative, the equation $|A| = -1$ has no solution. Final answer: a) $x = 3$ or $x = -3$ b) $x = 4$ or $x = -6$ c) $x = 5$ or $x = -1$ d) $x = 3$ or $x = -4$ e) No solution since absolute value cannot be negative.