inequalities: if you multiply or divide both sides by a negative number, reverse the inequality sign. This is the step students forget most often."},{"id":"block-2","content":"Example: solve $5 > 17 - 3x$. Subtract 17 from both sides to get $-12 > -3x$."},{"id":"block-3","content":"Now divide by -3 and reverse the sign to get $4 < x$. This is usually written as $x > 4$."}]},{"id":"card-11","type":"mcq","order":11,"options":[{"id":"a","text":"$$x \\le \\frac{4}{5}$","isCorrect":true},{"id":"b","text":"$$x \\ge \\frac{4}{5}$","isCorrect":false},{"id":"c","text":"$$x \\le -\\frac{4}{5}$","isCorrect":false},{"id":"d","text":"$$x \\ge -\\frac{4}{5}$","isCorrect":false}],"question":"Which is the correct solution to $7 - 2(x - 1) \\ge 3x + 5$?","explanation":"Expand first: $7 - 2(x - 1) = 9 - 2x$, so $9 - 2x \\ge 3x + 5$. Rearranging gives $4 \\ge 5x$, so $x \\le \\frac{4}{5}$. The common wrong answer is B, which comes from rewriting $4 \\ge 5x$ the wrong way round; dividing by positive 5 does not flip the sign. Hint: Expand the bracket before collecting the $x$ terms.","correctOptionId":"a"},{"id":"card-12","type":"flashcard","cards":[{"id":"fc-1","back":"Uses $<$ or $>$, so the endpoint is not included.","front":"Strict inequality"},{"id":"fc-2","back":"Uses $\\le$ or $\\ge$, so the endpoint is included.","front":"Inclusive inequality"},{"id":"fc-3","back":"Shows an endpoint that is not included in the solution set.","front":"Open circle"},{"id":"fc-4","back":"Shows an endpoint that is included in the solution set.","front":"Filled circle"},{"id":"fc-5","back":"A whole-number solution, including negatives, zero, and positive whole numbers.","front":"Integer solution"},{"id":"fc-6","back":"A statement with two bounds on the same variable, such as $-2 < x \\le 5$.","front":"Double inequality"}],"order":12},{"id":"card-13","type":"content","order":13,"title":"Solving double inequalities","blocks":[{"id":"block-1","content":"A double inequality has the variable between two bounds, such as $-2 < x \\le 5$. Both parts must be true at the same time."},{"id":"block-2","content":"When solving, do the same operation to all three parts. For $3 \\le 2y - 1 < 11$, add 1 everywhere to get $4 \\le 2y < 12$."},{"id":"block-3","content":"Now divide all three parts by 2 to get $2 \\le y < 6$. If $y$ is an integer, the possible values are $2, 3, 4, 5$."}]},{"id":"card-14","type":"fill_blank","order":14,"blanks":[{"id":"maxy","isMath":true,"correctAnswer":"5"}],"sentence":"If $y$ is an integer and $3 \\le 2y - 1 < 11$, then the greatest possible value of $y$ is {{blank:maxy}}."},{"id":"card-15","type":"multi-question","order":15,"questions":[{"id":"mq-1","options":[{"id":"a","text":"$$>$","isCorrect":false},{"id":"b","text":"$$\\ge$","isCorrect":true},{"id":"c","text":"$$\\le$","isCorrect":false}],"question":"Which symbol means \"greater than or equal to\"?","explanation":"The symbol $\\ge$ represents \"greater than or equal to,\" combining the greater than sign with a line representing equality."},{"id":"mq-2","options":[{"id":"a","text":"4","isCorrect":false},{"id":"b","text":"6","isCorrect":false},{"id":"c","text":"2","isCorrect":true}],"question":"Which number is a solution to $x < 4$?","explanation":"The inequality $x < 4$ includes all numbers strictly less than 4. Out of the choices, only 2 is less than 4."},{"id":"mq-3","options":[{"id":"a","text":"$$x > 3$","isCorrect":true},{"id":"b","text":"$$x \\ge 3$","isCorrect":false},{"id":"c","text":"$$x < 3$","isCorrect":false}],"question":"Solve $2x + 1 > 7$.","explanation":"Subtract 1 from both sides to get $2x > 6$. Then divide by 2 to find $x > 3$."},{"id":"mq-4","options":[{"id":"a","text":"$$x > 3$","isCorrect":true},{"id":"b","text":"$$x < 3$","isCorrect":false},{"id":"c","text":"$$x > -3$","isCorrect":false}],"question":"Solve $10 - 2x < 4$.","explanation":"Subtract 10 from both sides: $-2x < -6$. Dividing by $-2$ requires flipping the inequality sign, resulting in $x > 3$."},{"id":"mq-5","options":[{"id":"a","text":"3","isCorrect":false},{"id":"b","text":"4","isCorrect":true},{"id":"c",

Inequalities Lesson