Math

Rule for multiplying or dividing an inequality by a negative number

The direction of the inequality sign must be reversed (flipped).

Meaning of brackets $[ ]$ versus parentheses $( )$ in interval notation

Parentheses $( )$ denote excluded endpoints or infinite bounds ($-\infty$ or $\infty$), while brackets $[ ]$ denote included endpoints associated with closed circles on a number line.

Formula for the $x$-coordinate of the vertex of a parabola in standard form $y = ax^2 + bx + c$

$$x = -\frac{b}{2a}$$

Equivalent compound inequality for solving $|x| < a$ (where $a > 0$)

$$-a < x < a$$

Equivalent set of inequalities for solving $|x| \ge a$ (where $a > 0$)

$x \le -a$ or $x \ge a$

Vertex form equation of a quadratic function

$y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Method for clearing fractions in an algebraic inequality

Multiply every term on both sides by the least common multiple (LCM) of all denominators.

Effect of a negative leading coefficient ($a < 0$) on the graph of $y = ax^2 + bx + c$

The parabola opens downward and has a maximum point at its vertex.

Rule for multiplying or dividing an inequality by a negative number

The direction of the inequality sign must be reversed (flipped).

Meaning of brackets $[ ]$ versus parentheses $( )$ in interval notation

Parentheses $( )$ denote excluded endpoints or infinite bounds ($-\infty$ or $\infty$), while brackets $[ ]$ denote included endpoints associated with closed circles on a number line.

Formula for the $x$-coordinate of the vertex of a parabola in standard form $y = ax^2 + bx + c$

$x = -\frac{b}{2a}$\n\n

Equivalent compound inequality for solving $\|x\| < a$ (where $a > 0$)

$-a < x < a$\n\n

Equivalent set of inequalities for solving $\|x\| \ge a$ (where $a > 0$)

$x \le -a$ or $x \ge a$\n\n

Vertex form equation of a quadratic function

$y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.\n\n

Method for clearing fractions in an algebraic inequality

Multiply every term on both sides by the least common multiple (LCM) of all denominators.\n\n

Effect of a negative leading coefficient ($a < 0$) on the graph of $y = ax^2 + bx + c$

The parabola opens downward and has a maximum point at its vertex.\n\n

Type of circle used on a number line for strict inequalities ($<$ or $>$) versus inclusive inequalities ($\le$ or $\ge$)

An open circle is used for strict inequalities ($<$ or $>$) to denote exclusion, while a closed circle is used for inclusive inequalities ($\le$ or $\ge$) to denote inclusion.

Domain of any standard quadratic function $y = ax^2 + bx + c$

$$(-\infty, \infty)$$

Range of a quadratic function in vertex form $y = a(x - h)^2 + k$ when $a > 0$

$$[k, \infty)$$

Range of a quadratic function in vertex form $y = a(x - h)^2 + k$ when $a < 0$

$$(-\infty, k]$$

Equation for the axis of symmetry of a parabola given its vertex $(h, k)$

$$x = h$$

The two equations required to remove absolute value bars when solving $\|ax + b\| > c$ (where $c > 0$)

$ax + b > c$ or $ax + b < -c$

First step in solving a quadratic inequality such as $x^2 - x \ge 12$

Rearrange the inequality so that one side equals zero (e.g., $x^2 - x - 12 \ge 0$) before finding critical values.