1. Solve the first inequality: $10q - 12 < 48 + 5q$ Start by isolating $q$ on one side. $$10q - 12 < 48 + 5q$$ Subtract $5q$ from both sides: $$10q - \cancel{5q} - 12 < 48 + \cancel{5q}$$ $$5q - 12 < 48$$ Add 12 to both sides: $$5q - 12 + 12 < 48 + 12$$ $$5q < 60$$ Divide both sides by 5: $$\frac{5q}{\cancel{5}} < \frac{60}{\cancel{5}}$$ $$q < 12$$ **Answer:** $$q < 12$$ This means $q$ can be any number less than 12. --- 2. Solve the second inequality: $3g - 7 \geq 5g - 18$ Bring all $g$ terms to one side: $$3g - 7 \geq 5g - 18$$ Subtract $5g$ from both sides: $$3g - \cancel{5g} - 7 \geq \cancel{5g} - 18$$ $$-2g - 7 \geq -18$$ Add 7 to both sides: $$-2g - 7 + 7 \geq -18 + 7$$ $$-2g \geq -11$$ Divide both sides by $-2$ and reverse inequality sign because dividing by negative: $$\frac{-2g}{\cancel{-2}} \leq \frac{-11}{\cancel{-2}}$$ $$g \leq \frac{11}{2}$$ **Answer:** $$g \leq 5.5$$ --- 3. Solve the third inequality: $3 - 7h \leq 6 - 5h$ Bring $h$ terms to one side: $$3 - 7h \leq 6 - 5h$$ Add $7h$ to both sides: $$3 - \cancel{7h} + 7h \leq 6 - 5h + 7h$$ $$3 \leq 6 + 2h$$ Subtract 6 from both sides: $$3 - 6 \leq 6 - 6 + 2h$$ $$-3 \leq 2h$$ Divide both sides by 2: $$\frac{-3}{2} \leq h$$ or equivalently $$h \geq -1.5$$ **Answer:** $$h \geq -1.5$$ --- 4. Solve the fourth inequality: $3(h - 4) > 5(h - 10)$ Expand both sides: $$3h - 12 > 5h - 50$$ Bring $h$ terms to one side: $$3h - 12 > 5h - 50$$ Subtract $5h$ from both sides: $$3h - \cancel{5h} - 12 > \cancel{5h} - 50$$ $$-2h - 12 > -50$$ Add 12 to both sides: $$-2h - 12 + 12 > -50 + 12$$ $$-2h > -38$$ Divide both sides by $-2$ and reverse inequality sign: $$\frac{-2h}{\cancel{-2}} < \frac{-38}{\cancel{-2}}$$ $$h < 19$$ **Answer:** $$h < 19$$ --- 5. Solve the fifth inequality: $2(y - 7) + 6 \leq 5(y + 3) + 21$ Expand both sides: $$2y - 14 + 6 \leq 5y + 15 + 21$$ Simplify: $$2y - 8 \leq 5y + 36$$ Bring $y$ terms to one side: $$2y - 8 \leq 5y + 36$$ Subtract $5y$ from both sides: $$2y - \cancel{5y} - 8 \leq \cancel{5y} + 36$$ $$-3y - 8 \leq 36$$ Add 8 to both sides: $$-3y - 8 + 8 \leq 36 + 8$$ $$-3y \leq 44$$ Divide both sides by $-3$ and reverse inequality sign: $$\frac{-3y}{\cancel{-3}} \geq \frac{44}{\cancel{-3}}$$ $$y \geq -\frac{44}{3}$$ **Answer:** $$y \geq -14.67$$ --- 6. Solve the sixth inequality: $6(n - 4) - 2(n + 1) < 3(n + 7) + 1$ Expand: $$6n - 24 - 2n - 2 < 3n + 21 + 1$$ Simplify: $$4n - 26 < 3n + 22$$ Bring $n$ terms to one side: $$4n - 26 < 3n + 22$$ Subtract $3n$ from both sides: $$4n - \cancel{3n} - 26 < \cancel{3n} + 22$$ $$n - 26 < 22$$ Add 26 to both sides: $$n - 26 + 26 < 22 + 26$$ $$n < 48$$ **Answer:** $$n < 48$$ --- 7. Solve the seventh inequality: $5(2r - 3) - 2(4r - 5) \geq 8(r + 1)$ Expand: $$10r - 15 - 8r + 10 \geq 8r + 8$$ Simplify: $$2r - 5 \geq 8r + 8$$ Bring $r$ terms to one side: $$2r - 5 \geq 8r + 8$$ Subtract $8r$ from both sides: $$2r - \cancel{8r} - 5 \geq \cancel{8r} + 8$$ $$-6r - 5 \geq 8$$ Add 5 to both sides: $$-6r - 5 + 5 \geq 8 + 5$$ $$-6r \geq 13$$ Divide both sides by $-6$ and reverse inequality sign: $$\frac{-6r}{\cancel{-6}} \leq \frac{13}{\cancel{-6}}$$ $$r \leq -\frac{13}{6}$$ **Answer:** $$r \leq -2.17$$ --- 8. Solve the eighth inequality: $\frac{2e + 1}{9} > 7 - 6e$ Multiply both sides by 9: $$2e + 1 > 9(7 - 6e)$$ $$2e + 1 > 63 - 54e$$ Bring $e$ terms to one side: $$2e + 1 > 63 - 54e$$ Add $54e$ to both sides: $$2e + 54e + 1 > 63$$ $$56e + 1 > 63$$ Subtract 1 from both sides: $$56e > 62$$ Divide both sides by 56: $$e > \frac{62}{56} = \frac{31}{28} \approx 1.11$$ **Answer:** $$e > 1.11$$ --- 9. Solve the ninth inequality: $2t - \frac{2t + 1}{3} > 12$ Multiply both sides by 3 to clear denominator: $$3(2t) - (2t + 1) > 36$$ $$6t - 2t - 1 > 36$$ Simplify: $$4t - 1 > 36$$ Add 1 to both sides: $$4t > 37$$ Divide both sides by 4: $$t > \frac{37}{4} = 9.25$$ **Answer:** $$t > 9.25$$ --- 10. Solve the tenth inequality: $\frac{2}{3} - \frac{2t + 1}{9} > 12$ Multiply both sides by 9: $$9 \times \frac{2}{3} - (2t + 1) > 108$$ $$6 - 2t - 1 > 108$$ Simplify: $$5 - 2t > 108$$ Subtract 5 from both sides: $$-2t > 103$$ Divide both sides by $-2$ and reverse inequality sign: $$t < -\frac{103}{2} = -51.5$$ **Answer:** $$t < -51.5$$ --- 11. Solve the eleventh inequality: $\frac{2}{7} - \frac{2t + 1}{9} > 12$ Multiply both sides by 63 (LCM of 7 and 9): $$63 \times \frac{2}{7} - 63 \times \frac{2t + 1}{9} > 63 \times 12$$ Calculate: $$9 \times 2 - 7(2t + 1) > 756$$ $$18 - 14t - 7 > 756$$ Simplify: $$11 - 14t > 756$$ Subtract 11 from both sides: $$-14t > 745$$ Divide both sides by $-14$ and reverse inequality sign: $$t < -\frac{745}{14} \approx -53.21$$ **Answer:** $$t < -53.21$$