Absolute Value Equations 8Baf21

1. Solve $|8 + p| = 2p - 3$. - The absolute value equation $|A| = B$ implies $B \geq 0$ and $A = B$ or $A = -B$. - Here, $B = 2p - 3$, so $2p - 3 \geq 0 \Rightarrow p \geq \frac{3}{2}$. Case 1: $8 + p = 2p - 3$ $$8 + p = 2p - 3$$ $$8 + p - 2p = -3$$ $$8 - p = -3$$ $$-p = -11$$ $$p = 11$$ Check $p=11 \geq \frac{3}{2}$ valid. Case 2: $8 + p = -(2p - 3)$ $$8 + p = -2p + 3$$ $$8 + p + 2p = 3$$ $$8 + 3p = 3$$ $$3p = 3 - 8 = -5$$ $$p = -\frac{5}{3}$$ Check $p = -\frac{5}{3} \geq \frac{3}{2}$? No, discard. Final solution: $p = 11$. 2. Solve $|4w - 1| = 5w + 37$. - $5w + 37 \geq 0 \Rightarrow w \geq -\frac{37}{5} = -7.4$. Case 1: $4w - 1 = 5w + 37$ $$4w - 1 = 5w + 37$$ $$4w - 5w = 37 + 1$$ $$-w = 38$$ $$w = -38$$ Check $w = -38 \geq -7.4$? No, discard. Case 2: $4w - 1 = -(5w + 37)$ $$4w - 1 = -5w - 37$$ $$4w + 5w = -37 + 1$$ $$9w = -36$$ $$w = -4$$ Check $w = -4 \geq -7.4$ valid. Final solution: $w = -4$. 3. Solve $4|2y - 7| + 13 = 9$. $$4|2y - 7| = 9 - 13 = -4$$ Since absolute value is always $\geq 0$, $4|2y - 7| \geq 0$, cannot equal $-4$. No solution. 4. Solve $-2|7 - 3y| - 6 = -14$. $$-2|7 - 3y| = -14 + 6 = -8$$ $$|7 - 3y| = \frac{-8}{-2} = 4$$ So $|7 - 3y| = 4$. Case 1: $7 - 3y = 4$ $$7 - 3y = 4$$ $$-3y = 4 - 7 = -3$$ $$y = 1$$ Case 2: $7 - 3y = -4$ $$7 - 3y = -4$$ $$-3y = -4 - 7 = -11$$ $$y = \frac{11}{3}$$ Final solutions: $y = 1$ or $y = \frac{11}{3}$. 5. Solve $2|4 - n| = -3n$. - The left side $2|4 - n| \geq 0$. - The right side $-3n$ must be $\geq 0$ for equality, so $-3n \geq 0 \Rightarrow n \leq 0$. Case 1: $4 - n = \frac{-3n}{2}$ Multiply both sides by 2: $$2(4 - n) = -3n$$ $$8 - 2n = -3n$$ $$8 = -3n + 2n = -n$$ $$n = -8$$ Check $n = -8 \leq 0$ valid. Case 2: $4 - n = -\frac{-3n}{2} = \frac{3n}{2}$ Multiply both sides by 2: $$2(4 - n) = 3n$$ $$8 - 2n = 3n$$ $$8 = 3n + 2n = 5n$$ $$n = \frac{8}{5} = 1.6$$ Check $n = 1.6 \leq 0$? No, discard. Final solution: $n = -8$. 6. Solve $5 - 3|2 + 2w| = -7$. $$-3|2 + 2w| = -7 - 5 = -12$$ $$|2 + 2w| = \frac{-12}{-3} = 4$$ So $|2 + 2w| = 4$. Case 1: $2 + 2w = 4$ $$2w = 4 - 2 = 2$$ $$w = 1$$ Case 2: $2 + 2w = -4$ $$2w = -4 - 2 = -6$$ $$w = -3$$ Final solutions: $w = 1$ or $w = -3$. 7. Solve $5|2r + 3| - 5 = 0$. $$5|2r + 3| = 5$$ $$|2r + 3| = 1$$ Case 1: $2r + 3 = 1$ $$2r = 1 - 3 = -2$$ $$r = -1$$ Case 2: $2r + 3 = -1$ $$2r = -1 - 3 = -4$$ $$r = -2$$ Final solutions: $r = -1$ or $r = -2$. 8. Solve $3 - 5|2d - | = 4$. The expression $|2d - |$ is incomplete or malformed. Cannot solve due to incomplete problem statement. Final answers: 1. $p = 11$ 2. $w = -4$ 3. No solution 4. $y = 1$ or $y = \frac{11}{3}$ 5. $n = -8$ 6. $w = 1$ or $w = -3$ 7. $r = -1$ or $r = -2$ 8. Incomplete problem, no solution.