9000033307
Level:
B
Find the solution set of the following inequality.
\[
\frac{4}
{x^{2} - x - 6}\leq 0
\]
\((-2;3)\)
\(\mathbb{R}\)
\((-\infty ;-2)\cup (3;\infty )\)
\((-3;2)\)
Rational equations and inequalities
9000033308
Level:
C
Find the solution set of the following inequality.
\[
\frac{x^{2} + x + 2}
{x^{2} + 4x + 3}\geq 0
\]
\((-\infty ;-3)\cup (-1;\infty )\)
\((1;3)\)
\((-\infty ;1)\cup (3;\infty )\)
\((-3;-1)\)
9000033301
Level:
A
Find the solution set of the following equation.
\[
\frac{3x + 6}
{2 - x} = 0
\]
\(\{ - 2\}\)
\(\emptyset \)
\(\{2\}\)
\(\mathbb{R}\setminus \{2\}\)
9000033302
Level:
A
Find the solution set of the following equation.
\[
\frac{4x - 2}
{2x - 1} = 2
\]
\(\mathbb{R}\setminus \left \{\frac{1}
{2}\right \}\)
\(\mathbb{R}\)
\(\{2\}\)
\(\emptyset \)
9000025807
Level:
C
In the following list identify a true statement on the function
\(f\).
\[
f(x) = \frac{-2(3x + 1)}
{(2x + 3)(2 - x)}
\]
\(f(x) > 0 \iff x\in \left (-\frac{3}
{2};-\frac{1}
{3}\right )\cup (2;\infty )\)
\(f(x) > 0 \iff x\in \left (-\infty ;-\frac{3}
{2}\right )\cup \left (-\frac{1}
{3};2\right )\)
\(f(x) > 0 \iff x\in \left (-\frac{3}
{2};2\right )\)
\(f(x) > 0 \iff x\in \left (-\infty ;-\frac{3}
{2}\right )\cup (2;\infty )\)
9000026106
Level:
B
Find the solution set of the following inequality.
\[
\frac{x + 3}
{x - 1} > 1
\]
\(\left (1;\infty \right )\)
\(\left (-\infty ;1\right )\)
\(\left (-\infty ;3\right ] \)
\(\left [ 3;\infty \right )\)
9000026108
Level:
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\(2\leq \frac{x+3}
{x} \)
\(2\geq \frac{3}
{x}\)
\(2\geq \frac{x+3}
{x} \)
\(2\leq \frac{3}
{x}\)
9000024106
Level:
A
Identify the optimal first step to solve the following equation. The
operation is intended to be used on both sides of the equation. Assume
\(x\neq 1\) and
\(x\neq 2\).
\[
\frac{1}
{x - 1} = \frac{2}
{x - 2}
\]
multiply by \((x - 1)\cdot (x - 2)\)
multiply by \((x - 1)\)
multiply by \((x - 2)\)
multiply by \((x + 1)\)
multiply by \((x + 2)\)
multiply by \((x - 1)\cdot (x + 2)\)
9000024109
Level:
A
Identify the optimal first step to solve the following equation. The operation is
intended to be used on both sides of the equation.
\[
\frac{2x + 1}
{x - 1} + \frac{x + 1}
{x - 1} = \frac{11}
{2}
\]
multiply by \(2(x - 1)\),
assuming \(x\neq 1\)
multiply by \((2x + 1)\),
assuming \(x\neq -\frac{1}
{2}\)
multiply by \((x + 1)\),
assuming \(x\neq - 1\)
multiply by \(\frac{1}
{2x+1}\),
assuming \(x\neq -\frac{1}
{2}\)
multiply by \(\frac{1}
{x+1}\),
assuming \(x\neq - 1\)
multiply by \(2(2x + 1)(x + 1)\),
assuming \(x\neq -\frac{1}
{2}\)
and \(x\neq - 1\)
9000024105
Level:
A
Identify the optimal first step to solve the following equation. The operation is
intended to be used on both sides of the equation.
\[
\frac{4 + x}
{x + 1} = \frac{x - 3}
{x + 2}
\]
multiply by \((x + 2)\cdot (x + 1)\),
assuming \(x\neq - 2\)
and \(x\neq - 1\)
multiply by \((4 + x)\cdot (x - 3)\),
assuming \(x\neq - 4\)
and \(x\neq 3\)
multiply by \((4 + x)\cdot (x + 1)\),
assuming \(x\neq - 4\)
and \(x\neq - 1\)
multiply by \((x - 3)\cdot (x + 2)\),
assuming \(x\neq 3\)
and \(x\neq - 2\)
multiply by \((x - 3)\),
assuming \(x\neq 3\)
multiply by \((4 + x)\),
assuming \(x\neq - 4\)