Radical equations and inequalities

9000023810

Level: 
A
Denote by \(x_{1}\) the solution of the equation \[ \sqrt{6 - 2x} = -x - 1 \] and by \(x_{2}\) the solution of the equation \[ \sqrt{2x + 6} = 9 - x. \] Identify a correct statement about \(x_{1}\) and \(x_{2}\).
\(|x_{1}| = |x_{2}|\)
\(|x_{1}| < |x_{2}|\)
\(|x_{1}| > |x_{2}|\)
\(5|x_{1}| = |x_{2}|\)

9000023709

Level: 
A
Identify a true statement which concerns the following pair of equations. \[ \begin{aligned} \sqrt{ 5 - x} & = 2 &\text{(1)} \\ \sqrt{x + 5} & = 4 &\text{(2)} \end{aligned} \]
The solution of (1) is smaller than the solution of (2).
The solutions of both equations are prime numbers.
The solution of (1) is bigger than the solution of (2).
The solution of (1) equals to the solution of (2).

9000020006

Level: 
A
Identify a true statement referring to the solution of the following equation. \[ \sqrt{3x - 8} = x - 6 \]
The equation has a unique solution and this solution is an odd number.
The equation has two solutions, the sum of these solutions is divisible by \(5\).
The equation has a unique solution and this solution is an even number.
The equation does not have a solution in \(\mathbb{R}\).

9000020007

Level: 
A
Identify a true statement referring to the solution of the following equation. \[ \sqrt{x^{2 } - 4} = x + 1 \]
The equation does not have a solution in \(\mathbb{R}\).
The equation has a unique negative solution.
The equation has a unique positive solution.
The equation has two solutions.

9000020008

Level: 
A
Identify a true statement referring to the solution of the following equation. \[ 6x - 13\sqrt{x} + 6 = 0 \] Hint: Use the substitution \(y = \sqrt{x}\).
The solutions \(x_{1}\) and \(x_{2}\) satisfy \(x_{1} = \frac{1} {x_{2}} \).
The equation has a unique solution \(x_{1}\). This solution satisfies \(x_{1} < 1\).
The equation has a unique solution \(x_{1}\). This solution satisfies \(x_{1} > 1\).
The equation does not have a solution in \(\mathbb{R}\).