Числовые неравенства

{"questions":[{"content":"$$14+2x>44$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$x>15$","$x<15$","$x>29$","$x<29$"],"placeholder":0,"answer":0}},"hints":["Перенесем слагаемое в правую часть, изменив его знак:<br />$2x>44−14$","$2x>30$<br />$x>15$."]}]}

{"questions":[{"content":"$$9+3x\\le 30$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$x\\le 7$","$x\\ge 7$","$x\\le 13$","$x\\ge 13$"],"placeholder":0,"answer":0}},"hints":["Перенесем число в правую часть:<br>$3x\\le 30-9$.","$3x\\le 21$.<br>Разделим обе части на $3$:<br>$x\\le 7$."]},{"content":"$$5x-12>28$$[[fill_choice_big-2]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$x>8$","$x>\\frac{16}{5}$","$x<8$","$x>\\frac{8}{5}$"],"placeholder":0,"answer":0}},"hints":["Перенесем $-12$ в правую часть:<br>$5x>28+12$.","$5x>40$.<br>Разделим обе части на $5$:<br>$x>8$."]},{"content":"$$42-6x\\ge 18$$[[fill_choice_big-3]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$x\\le 4$","$x\\ge 4$","$x\\le -4$","$x\\ge -4$"],"placeholder":0,"answer":0}},"hints":["Перенесем $42$ в правую часть:<br>$-6x\\ge 18-42$.","$-6x\\ge -24$.<br>Разделим на отрицательное число $-6$ и поменяем знак неравенства:<br>$x\\le 4$."]},{"content":"$$\\frac{x}{4}-3<2$$[[fill_choice_big-4]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$x<20$","$x>20$","$x<8$","$x>8$"],"placeholder":0,"answer":0}},"hints":["Прибавим $3$ к обеим частям:<br>$\\frac{x}{4}<2+3$.","$\\frac{x}{4}<5$.<br>Умножим обе части на $4$:<br>$x<20$."]},{"content":"$$7-\\frac{x}{2}\\le -1$$[[fill_choice_big-5]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$x\\ge 16$","$x\\le 16$","$x\\ge -16$","$x\\le -16$"],"placeholder":0,"answer":0}},"hints":["Перенесем $7$ в правую часть:<br>$-\\frac{x}{2}\\le -1-7$.","$-\\frac{x}{2}\\le -8$.<br>Умножим на $-2$ и поменяем знак неравенства:<br>$x\\ge 16$."]},{"content":"$$2(3x-5)\\ge 4x+6$$[[fill_choice_big-6]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$x\\ge 8$","$x\\le 8$","$x\\ge 4$","$x\\le 4$"],"placeholder":0,"answer":0}},"hints":["Раскроем скобки:<br>$6x-10\\ge 4x+6$.","Перенесем $4x$ влево, числа вправо:<br>$2x\\ge 16$.<br>Разделим на $2$:<br>$x\\ge 8$."]},{"content":"$$3(x+4)<2x+1$$[[fill_choice_big-7]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$x<-11$","$x>11$","$x<-5$","$x>-5$"],"placeholder":0,"answer":0}},"hints":["Раскроем скобки:<br>$3x+12<2x+1$.","Перенесем $2x$ влево, $12$ вправо:<br>$x<1-12$.<br>$x<-11$."]},{"content":"$$\\frac{5x-1}{3}>4$$[[fill_choice_big-8]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$x>\\frac{13}{5}$","$x<\\frac{13}{5}$","$x>\\frac{5}{13}$","$x>\\frac{11}{5}$"],"placeholder":0,"answer":0}},"hints":["Умножим обе части на $3$:<br>$5x-1>12$.","$5x>13$.<br>Разделим на $5$:<br>$x>\\frac{13}{5}$."]},{"content":"$$-4(x-2)\\le 8$$[[fill_choice_big-9]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$x\\ge 0$","$x\\le 0$","$x\\ge -4$","$x\\le -4$"],"placeholder":0,"answer":0}},"hints":["Раскроем скобки:<br>$-4x+8\\le 8$.","Вычтем $8$ из обеих частей:<br>$-4x\\le 0$.<br>Разделим на $-4$ и поменяем знак неравенства:<br>$x\\ge 0$."]},{"content":"$$\\frac{2x+5}{5}\\ge \\frac{x-1}{2}$$[[fill_choice_big-10]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$x\\ge -15$","$x\\le -15$","$x\\ge 15$","$x\\le 15$"],"placeholder":0,"answer":0}},"hints":["Умножим обе части на $10$ (НОК знаменателей):<br>$2(2x+5)\\ge 5(x-1)$.","Раскроем скобки:<br>$4x+10\\ge 5x-5$.<br>Перенесем $4x$ вправо, числа влево:<br>$15\\ge x$.<br>Это равносильно $x\\le 15$."]}],"mix":1}

{"questions":[{"content":"$$8x-15\\le 9$$[[fill_choice_big-11]]","widgets":{"fill_choice_big-11":{"type":"fill_choice_big","options":["$x\\le 3$","$x\\ge 3$","$x\\le \\frac{3}{8}$","$x\\ge \\frac{3}{8}$"],"placeholder":0,"answer":0}},"hints":["Перенесем $-15$ в правую часть:<br>$8x\\le 9+15$.","$8x\\le 24$.<br>Разделим обе части на $8$:<br>$x\\le 3$."]},{"content":"$$11-3x>2$$[[fill_choice_big-12]]","widgets":{"fill_choice_big-12":{"type":"fill_choice_big","options":["$x<3$","$x>3$","$x<\\frac{1}{3}$","$x>\\frac{1}{3}$"],"placeholder":0,"answer":0}},"hints":["Перенесем $11$ в правую часть:<br>$-3x>2-11$.","$-3x>-9$.<br>Разделим на $-3$ и поменяем знак неравенства:<br>$x<3$."]},{"content":"$$\\frac{x}{6}+5\\ge 2$$[[fill_choice_big-13]]","widgets":{"fill_choice_big-13":{"type":"fill_choice_big","options":["$x\\ge -18$","$x\\le -18$","$x\\ge 18$","$x\\le 18$"],"placeholder":0,"answer":0}},"hints":["Вычтем $5$ из обеих частей:<br>$\\frac{x}{6}\\ge 2-5$.","$\\frac{x}{6}\\ge -3$.<br>Умножим обе части на $6$:<br>$x\\ge -18$."]},{"content":"$$5-\\frac{2x}{3}<1$$[[fill_choice_big-14]]","widgets":{"fill_choice_big-14":{"type":"fill_choice_big","options":["$x>6$","$x<6$","$x>\\frac{3}{2}$","$x<\\frac{3}{2}$"],"placeholder":0,"answer":0}},"hints":["Перенесем $5$ в правую часть:<br>$-\\frac{2x}{3}<1-5$.","$-\\frac{2x}{3}<-4$.<br>Умножим на $-\\frac{3}{2}$ и поменяем знак неравенства:<br>$x>6$."]},{"content":"$$-7x+4\\ge 25$$[[fill_choice_big-15]]","widgets":{"fill_choice_big-15":{"type":"fill_choice_big","options":["$x\\le -3$","$x\\ge -3$","$x\\le 3$","$x\\ge 3$"],"placeholder":0,"answer":0}},"hints":["Перенесем $4$ в правую часть:<br>$-7x\\ge 25-4$.","$-7x\\ge 21$.<br>Разделим на $-7$ и поменяем знак неравенства:<br>$x\\le -3$."]},{"content":"$$2(4x-1)<3x+14$$[[fill_choice_big-16]]","widgets":{"fill_choice_big-16":{"type":"fill_choice_big","options":["$x<\\frac{16}{5}$","$x>\\frac{16}{5}$","$x<\\frac{12}{5}$","$x>\\frac{12}{5}$"],"placeholder":0,"answer":0}},"hints":["Раскроем скобки:<br>$8x-2<3x+14$.","Перенесем $3x$ влево, числа вправо:<br>$5x<16$.<br>Разделим на $5$:<br>$x<\\frac{16}{5}$."]},{"content":"$$6(x-3)\\ge 2(2x+9)$$[[fill_choice_big-17]]","widgets":{"fill_choice_big-17":{"type":"fill_choice_big","options":["$x\\ge 9$","$x\\le 9$","$x\\ge 3$","$x\\le 3$"],"placeholder":0,"answer":0}},"hints":["Раскроем скобки:<br>$6x-18\\ge 4x+18$.","Перенесем $4x$ влево, числа вправо:<br>$2x\\ge 36$.<br>Разделим на $2$:<br>$x\\ge 18$."]},{"content":"$$4(x+2)-3\\le 2x+9$$[[fill_choice_big-18]]","widgets":{"fill_choice_big-18":{"type":"fill_choice_big","options":["$x\\le 2$","$x\\ge 2$","$x\\le 6$","$x\\ge 6$"],"placeholder":0,"answer":0}},"hints":["Раскроем скобки и приведем подобные:<br>$4x+8-3\\le 2x+9$, то есть $4x+5\\le 2x+9$.","Перенесем $2x$ влево, числа вправо:<br>$2x\\le 4$.<br>Разделим на $2$:<br>$x\\le 2$."]},{"content":"$$\\frac{3x-7}{2}>5$$[[fill_choice_big-19]]","widgets":{"fill_choice_big-19":{"type":"fill_choice_big","options":["$x>\\frac{17}{3}$","$x<\\frac{17}{3}$","$x>\\frac{7}{3}$","$x<\\frac{7}{3}$"],"placeholder":0,"answer":0}},"hints":["Умножим обе части на $2$:<br>$3x-7>10$.","$3x>17$.<br>Разделим на $3$:<br>$x>\\frac{17}{3}$."]},{"content":"$$\\frac{x+1}{3}-2\\le 0$$[[fill_choice_big-20]]","widgets":{"fill_choice_big-20":{"type":"fill_choice_big","options":["$x\\le 5$","$x\\ge 5$","$x\\le -5$","$x\\ge -5$"],"placeholder":0,"answer":0}},"hints":["Прибавим $2$ к обеим частям:<br>$\\frac{x+1}{3}\\le 2$.","Умножим на $3$:<br>$x+1\\le 6$.<br>Вычтем $1$:<br>$x\\le 5$."]},{"content":"$$9-2(3x-4)>1$$[[fill_choice_big-21]]","widgets":{"fill_choice_big-21":{"type":"fill_choice_big","options":["$x<\\frac{8}{3}$","$x>\\frac{8}{3}$","$x<\\frac{10}{3}$","$x>\\frac{10}{3}$"],"placeholder":0,"answer":0}},"hints":["Раскроем скобки:<br>$9-(6x-8)>1$, то есть $9-6x+8>1$.","$17-6x>1$.<br>Перенесем $17$ вправо:<br>$-6x>-16$.<br>Разделим на $-6$ и поменяем знак:<br>$x<\\frac{8}{3}$."]},{"content":"$$-\\frac{x}{5}+6\\ge 4$$[[fill_choice_big-22]]","widgets":{"fill_choice_big-22":{"type":"fill_choice_big","options":["$x\\le 10$","$x\\ge 10$","$x\\le -10$","$x\\ge -10$"],"placeholder":0,"answer":0}},"hints":["Вычтем $6$ из обеих частей:<br>$-\\frac{x}{5}\\ge 4-6$.","$-\\frac{x}{5}\\ge -2$.<br>Умножим на $-5$ и поменяем знак неравенства:<br>$x\\le 10$."]},{"content":"$$\\frac{7x+2}{4}<\\frac{3x-1}{2}$$[[fill_choice_big-23]]","widgets":{"fill_choice_big-23":{"type":"fill_choice_big","options":["$x>-4$","$x<-4$","$x>\\frac{2}{7}$","$x<\\frac{2}{7}$"],"placeholder":0,"answer":1}},"hints":["Умножим обе части на $4$ (НОК знаменателей):<br>$7x+2<2(3x-1)$.","Раскроем скобки:<br>$7x+2<6x-2$.<br>Перенесем $6x$ влево:<br>$x+2<-2$.<br>$x<-4$."]},{"content":"$$3(2x+1)-5\\ge x+10$$[[fill_choice_big-24]]","widgets":{"fill_choice_big-24":{"type":"fill_choice_big","options":["$x\\ge 6$","$x\\le 6$","$x\\ge 3$","$x\\le 3$"],"placeholder":0,"answer":0}},"hints":["Раскроем скобки и упростим:<br>$6x+3-5\\ge x+10$, то есть $6x-2\\ge x+10$.","Перенесем $x$ влево, числа вправо:<br>$5x\\ge 12$.<br>Разделим на $5$:<br>$x\\ge \\frac{12}{5}$."]},{"content":"$$\\frac{2x-3}{5}+\\frac{x+2}{10}\\le 1$$[[fill_choice_big-25]]","widgets":{"fill_choice_big-25":{"type":"fill_choice_big","options":["$x\\le 3$","$x\\ge 3$","$x\\le -3$","$x\\ge -3$"],"placeholder":0,"answer":0}},"hints":["Приведем к общему знаменателю $10$:<br>$\\frac{2(2x-3)}{10}+\\frac{x+2}{10}\\le 1$.","Сложим дроби:<br>$\\frac{4x-6+x+2}{10}\\le 1$, то есть $\\frac{5x-4}{10}\\le 1$.<br>Умножим на $10$:<br>$5x-4\\le 10$.<br>$5x\\le 14$.<br>$x\\le \\frac{14}{5}$."]}]}