Rational equations and inequalities

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 \(x\neq - 1\)

9000022804

Level: 
B
Establish the values of the real parameter \(t\) which ensure that the following expression is nonpositive. \[ \frac{2} {2t^{2} + t - 1} \]
\(\left (-1; \frac{1} {2}\right )\)
\(\left [ -\frac{1} {2};1\right ] \)
\(\left [ -1; \frac{1} {2}\right ] \)
\(\left (-\frac{1} {2};1\right )\)

9000021804

Level: 
B
Solve the following inequality. \[ \frac{1} {x - 3}\leq \frac{1} {2 - x} \]
\(x\in (-\infty ;2)\cup \left [ \frac{5} {2};3\right )\)
\(x\in (-\infty ;2)\cup \left [ \frac{5} {3};2\right ] \)
\(x\in \left (-\infty ; \frac{5} {2}\right ] \cup \left (3;\infty \right )\)
\(x\in \left [ \frac{5} {2};\infty \right )\)

9000021810

Level: 
B
Find all the values of \(x\) for which the following expression takes on values smaller than or equal to \(1\). \[ \frac{x + 1} {x - 1} - \frac{1} {x + 1} \]
\(x\in (-\infty ;-3] \cup (-1;1)\)
\(x\in (-\infty ;-3] \)
\(x\in (-\infty ;-1)\cup (-1;1)\cup (1;\infty )\)
\(x\in [ - 3;-1)\)