Graduate Record Examinations (GRE) Quantitative Reasoning: Algebra Syllabus

Solve $x^{2} = 49$.

$x = \pm 7$. Taking a square root introduces both positive and negative roots.

What is the key rule when multiplying or dividing an inequality by a negative number?

You must reverse the direction of the inequality sign.

Solve the inequality $-2x + 3 < 7$.

$-2x < 4$, then divide by $-2$ and flip: $x > -2$.

What is the definition of absolute value $|x|$?

$|x| = x$ if $x \geq 0$ and $|x| = -x$ if $x < 0$; it is the distance of $x$ from $0$ on the number line, always nonnegative.

How do you solve the absolute value equation $|x| = a$ for $a > 0$?

Split into two cases: $x = a$ or $x = -a$.

How do you rewrite the inequality $|x| < a$ (with $a > 0$) without absolute value?

$-a < x < a$. The solution is the interval between $-a$ and $a$.

How do you rewrite the inequality $|x| > a$ (with $a > 0$) without absolute value?

$x < -a$ or $x > a$. The solution is the two outer regions.

Solve $|2x - 1| = 5$.

$2x - 1 = 5$ gives $x = 3$; $2x - 1 = -5$ gives $x = -2$. So $x = 3$ or $x = -2$.

What is the general process for solving a quadratic inequality like $x^{2} - x - 6 > 0$?

Find the roots ($x=3, x=-2$), mark them on a number line, and test the sign of the expression in each interval to determine where the inequality holds (here $x < -2$ or $x > 3$).

Why must you account for the sign of the denominator when solving a rational inequality such as $\dfrac{x-1}{x+2} \geq 0$?

The expression can change sign at zeros of the numerator and at undefined points (zeros of the denominator), and the denominator's value cannot be zero, so those critical points partition the number line into test intervals.

For $\dfrac{x-1}{x+2} \geq 0$, what are the critical values and excluded value?

Critical values are $x = 1$ (numerator zero) and $x = -2$ (denominator zero); $x = -2$ is excluded because the expression is undefined there.

In function notation, what does $f(3)$ mean?

It is the output value of the function $f$ when the input $x = 3$; substitute $3$ for $x$ in the rule for $f$.

If $f(x) = 2x^{2} - x + 1$, evaluate $f(-2)$.

$f(-2) = 2(4) - (-2) + 1 = 8 + 2 + 1 = 11$.

What is the slope-intercept form of a line and what does each parameter represent?

$y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

State the slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$.

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$.

What is the point-slope form of a line through $(x_1, y_1)$ with slope $m$?

$y - y_1 = m(x - x_1)$.

How are the slopes of parallel and perpendicular lines related?

Parallel lines have equal slopes ($m_1 = m_2$); perpendicular lines have slopes that are negative reciprocals ($m_1 m_2 = -1$).

What is the slope of a horizontal line versus a vertical line?

A horizontal line has slope $0$ (equation $y = c$); a vertical line has undefined slope (equation $x = c$).

What is the shape and key feature of the graph of $y = x^{2}$?

A parabola opening upward with vertex at the origin $(0,0)$, symmetric about the $y$-axis.

What does the graph of the absolute value function $y = |x|$ look like?

A V-shape with its vertex at the origin, slope $-1$ for $x<0$ and slope $+1$ for $x>0$.

What is the basic shape of the graph of $y = \sqrt{x}$?

Half of a sideways parabola starting at the origin, defined only for $x \geq 0$, increasing and curving to the right.

How does $y = f(x) + k$ transform the graph of $y = f(x)$?

It shifts the graph vertically: up by $k$ if $k > 0$, down if $k < 0$.

How does $y = f(x - h)$ transform the graph of $y = f(x)$?

It shifts the graph horizontally to the right by $h$ (left if $h$ is negative).

Translate into an equation: 'Five less than twice a number $n$ is 17.'

$2n - 5 = 17$.

Translate into an expression: 'the sum of a number and 8, all divided by 3.'

$\dfrac{x + 8}{3}$.

What is the standard distance-rate-time relationship used in word problems?

$d = rt$ (distance equals rate times time).

Set up the equation for a mixture problem: how many liters of a $20\%$ acid solution must be added to $10$ L of a $50\%$ solution to get a $30\%$ solution?

$0.20x + 0.50(10) = 0.30(x + 10)$, where $x$ is the liters of $20\%$ solution added.

Set up an age problem: Sara is 4 years older than Tom, and in 6 years she will be twice his current age. Let Tom's age be $t$.

Sara now $= t + 4$; in 6 years $t + 4 + 6 = 2t$, i.e. $t + 10 = 2t$, giving $t = 10$.

In a money problem, if you have $x$ dimes and $y$ quarters totaling $\$3.40$, what equation models the value in dollars?

$0.10x + 0.25y = 3.40$ (or $10x + 25y = 340$ in cents).

State the formula for the $n$-th term of an arithmetic sequence with first term $a_1$ and common difference $d$.

$a_n = a_1 + (n-1)d$.

State the formula for the $n$-th term of a geometric sequence with first term $a_1$ and common ratio $r$.

$a_n = a_1 \cdot r^{\,n-1}$.

How do you find the common difference of an arithmetic sequence and the common ratio of a geometric sequence?

Common difference $d = a_{n+1} - a_n$ (subtract consecutive terms); common ratio $r = \dfrac{a_{n+1}}{a_n}$ (divide consecutive terms).