d":"item-5","content":"Include or exclude the endpoints to match the inequality sign."}],"order":8,"instruction":"Put these steps for solving a quadratic inequality into the correct order.","correctOrder":["item-1","item-2","item-3","item-4","item-5"]},{"id":"card-9","type":"content","order":9,"title":"Rearrange before factorising","blocks":[{"id":"block-1","content":"Some inequalities are not already equal to zero on one side. Solve $5x+14>x^2$ by subtracting $5x+14$ from both sides to set one side of the inequality to zero:"},{"id":"block-2","content":"$$$0>x^2-5x-14$$"},{"id":"block-3","content":"We rewrite this as $x^2-5x-14<0$. Factorising then gives $(x-7)(x+2)<0$, which reveals the boundary values of $x=7$ and $x=-2$."},{"id":"block-4","content":"Since the coefficient of $x^2$ is positive, the less-than-zero region is found between the two roots. Therefore, the solution is: $$-2x^2$: $x$ is between {{blank:low_between1}} and {{blank:high_between1}}."},{"id":"card-11","type":"content","order":11,"title":"Special cases and non-monic quadratics","blocks":[{"id":"block-1","content":"If there is no $x$ term, it is still a quadratic inequality. To solve $x^2 \\le 64$, we rearrange the terms into standard form and factorise the expression into its linear components."},{"id":"block-2","content":"$$$x^2-64\\le 0 \\implies (x-8)(x+8)\\le 0$$"},{"id":"block-3","content":"The roots are $-8$ and $8$, so the solution is $-8\\le x\\le 8$. Do not forget the negative root, because squaring a negative number can still give a value less than or equal to $64$."},{"id":"block-4","content":"Non-monic quadratics use the same method. For example, given $6x^2+7x-3=(3x-1)(2x+3)$, the inequality $6x^2+7x-3<0$ is satisfied between the roots $x=-\\frac{3}{2}$ and $x=\\frac{1}{3}$."}]},{"id":"card-12","type":"mcq","order":12,"options":[{"id":"a","text":"$$-8\\le x\\le 8$","isCorrect":true},{"id":"b","text":"$$x\\le 8$","isCorrect":false},{"id":"c","text":"$$x\\ge -8$","isCorrect":false},{"id":"d","text":"$$x<-8$ or $x>8$","isCorrect":false}],"question":"Solve $x^2\\le 64$.","explanation":"Rewrite as $x^2-64\\le 0$ and use the roots $-8$ and $8$; an upward-opening quadratic is less than or equal to zero between the roots. The common wrong answer is $x\\le 8$, which forgets that negative values like $x=-5$ also satisfy the inequality.","correctOptionId":"a"},{"id":"card-13","type":"content","order":13,"title":"Stretch: listing integer solutions","blocks":[{"id":"block-1","content":"Sometimes the boundary values are not integers, so factorising is not convenient. For $x^2-6x+7<0$, use the quadratic formula on the related equation: $$x=\\frac{6\\pm\\sqrt{6^2-4\\cdot 1\\cdot 7}}{2}$$"},{"id":"block-2","content":"This simplifies to $x=3\\pm\\sqrt{2}$. Since the quadratic opens upwards and the inequality is less than zero, the full solution is $$3-\\sqrt{2}3$","isCorrect":false},{"id":"c","text":"$$x\\le -3$ or $x\\ge 3$","isCorrect":false}],"question":"Solve $x^2-9<0$.","explanation":"The boundary values are $x = \\pm 3$. For an upward-opening parabola, the inequality $x^2-9 < 0$ is satisfied in the interval between the roots."},{"id":"mq-2","options":[{"id":"a","text":"

Quadratic Inequalities Lesson

Quadratic Inequalities Lesson