Radical equations and inequalities

9000024802

Level: 
A
Consider the equation \[ \sqrt{x^{2 } - 2x + 1} = x + 2 \] and the equation which arises from this equation by squaring both sides of the equation, i.e. the equation \[ \left (\sqrt{x^{2 } - 2x + 1}\right )^{2} = (x + 2)^{2}. \] Identify a true statement.
Both equations are equivalent only if \(x\geq - 2\).
Both equations are equivalent.
Both equations are equivalent only if \(x\leq - 2\).
None of the above.

9000024803

Level: 
A
Removing radical in an equation by squaring both sides may enrich the set of solutions of this equation and checking the solutions of the new equation in the original equation may be necessary. Identify a correct conclusion in the particular case of the following equation. \[ -\sqrt{x^{2 } - 2x + 1} = x \]
If we look for the solution in the set \(\mathbb{R}^{-}\), then squaring both sides of the equation gives an equivalent equation. The checking of the solution is not necessary.
If we look for the solution in the set \(\mathbb{R}^{+}\), then squaring both sides of the equation gives an equivalent equation. The checking of the solution is not necessary.
If we look for the solution in the set \(\mathbb{R}\), then squaring both sides of the equation gives an equivalent equation. The checking of the solution is not necessary.
None of the above.

9000033702

Level: 
A
Find the domain of the following expression. \[ \sqrt{-x^{2 } + 7x - 12} -\frac{1} {x} \]
\([ 3;4] \)
\(\mathbb{R}\setminus \left \{0\right \}\)
\(\mathbb{R}\setminus \left \{0;3;4\right \}\)
\(\left (3;4\right )\)
\(\left (-\infty ;3\right )\cup \left (4;\infty \right )\)
\(\left (-\infty ;3] \cup [ 4;\infty \right )\)

9000034901

Level: 
A
Find the domain of the following expression. \[ \sqrt{\left (2x - 3 \right ) \left (3x + 1 \right )} \]
\(\left (-\infty ;-\frac{1} {3}\right ] \cup \left [ \frac{3} {2};\infty \right )\)
\(\left [ -\frac{1} {3}; \frac{3} {2}\right ] \)
\(\left (-\frac{1} {3}; \frac{3} {2}\right )\)
\(\left (-\infty ;-\frac{1} {3}\right )\cup \left (\frac{3} {2};\infty \right )\)

9000034903

Level: 
A
Find all \(x\in \mathbb{R}\) for which the following expression is undefined. \[ \sqrt{\left (3x + 4 \right ) \left (\frac{1} {5} - x\right )} \]
\(\left (-\infty ;-\frac{4} {3}\right )\cup \left (\frac{1} {5};\infty \right )\)
\(\left [ -\frac{4} {3}; \frac{1} {5}\right ] \)
\(\left (-\infty ;-\frac{4} {3}\right ] \cup \left [ \frac{1} {5};\infty \right )\)
\(\left (-\frac{4} {3}; \frac{1} {5}\right )\)