Rational equations and inequalities

2000005307

Level:

Use the graphs of the functions \(f(x)=x+3\) and \(g(x)=x-1\) to find the solution set of the following inequality. \[\frac{x+3}{x-1} \leq 0\]

\( x \in [ -3;1) \)

\( x \in [ -3;1] \)

\( x \in (1;+\infty) \)

\( x \in (-1;+\infty) \)

2000005308

Level:

Use the graphs of the functions \(f(x)=x-3\) and \(g(x)=4-x\) to find the solution set of the following inequality. \[\frac{x-3}{4-x} < 0\]

\( x \in (-\infty;3) \cup (4;+\infty) \)

\( x \in (3;4) \)

\( x \in (-\infty;3) \)

\( x \in (-\infty;0) \)

2010011901

Level:

Find the solution set of the given inequality. \[ \frac{|4x+1|}{x(x+3)} \leq 0 \]

\( (-3;0) \)

\( [ -3;0 ] \)

\( (-\infty;-3)\cup(0;\infty) \)

\(\left(-3;-\frac14 \right]\)

2010011902

Level:

Find the solution set of the given inequality. \[ \frac{|3x+4|}{2-x} > 1 \]

\( (-\infty;-3)\cup\left(-\frac12;2\right) \)

\( (2;\infty)\)

\( (-\infty;-3)\cup(2;\infty) \)

\((-3;-\frac12 ]\)

2010011903

Level:

Find the solution set of the given inequality. \[ \frac{x+1}{|3-x|} < 1 \]

\( (-\infty;1) \)

\( [ -1;3)\)

\([ 1,3)\cup(3;\infty) \)

\((-\infty;-1 ]\)

2110011904

Level:

Identify the picture that shows the correct solution set of the following inequality. In each picture, the set of points corresponding to the solution set is marked in red. \[ \frac{2} {x}\geq x-1 \]

9000025807

Level:

In the following list identify a true statement on the function \(f\). \[ f(x) = \frac{-2(3x + 1)} {(2x + 3)(2 - x)} \]

\(f(x) > 0 \iff x\in \left (-\frac{3} {2};-\frac{1} {3}\right )\cup (2;\infty )\)

\(f(x) > 0 \iff x\in \left (-\infty ;-\frac{3} {2}\right )\cup \left (-\frac{1} {3};2\right )\)

\(f(x) > 0 \iff x\in \left (-\frac{3} {2};2\right )\)

\(f(x) > 0 \iff x\in \left (-\infty ;-\frac{3} {2}\right )\cup (2;\infty )\)

9000033308

Level:

Find the solution set of the following inequality. \[ \frac{x^{2} + x + 2} {x^{2} + 4x + 3}\geq 0 \]

\((-\infty ;-3)\cup (-1;\infty )\)

\((1;3)\)

\((-\infty ;1)\cup (3;\infty )\)

\((-3;-1)\)

9100026109

Level:

Identify the picture that shows the correct solution set of the following inequality. In each picture, the set of points corresponding to the solution set is marked in red. \[ \frac{4} {x}\leq x \]