Solutions and Explanations
to Intermediate Algebra Questions in Sample 4
Full explanations of solutions for intermediate algebra questions in sample 4 are presented.
Solutions
-
(True or False) \(x^2\) and \(2x\) are like terms.
Solution
The statement "\(x^2\) and \(2x\) are like terms" is FALSE because the two terms do not have the same power of \(x\).
-
(True or False) \(x^{-3}\) and \(-3x\) are unlike terms.
Solution
The statement "\(x^{-3}\) and \(-3x\) are unlike terms" is TRUE because the two terms do not have the same power of \(x\).
-
(True or False) \(\frac{1}{x - 9} = 0\) for \(x = 9\).
Solution
Substitute \(x\) by 9 in the expression \(\frac{1}{x - 9}\).
\[
\frac{1}{x - 9} = \frac{1}{9 - 9} = \frac{1}{0} = \text{undefined}
\]
The statement "\(\frac{1}{x - 9} = 0\) for \(x = 9\)" is FALSE.
-
(True or False) The set of ordered pairs \(\{(0,0),(2,0),(3,0),(10,0)\}\) represents a function.
Solution
All values of the \(x\) coordinates are different and therefore the set of ordered pairs represents a function. The statement is TRUE.
-
(True or False) \(|a - b| = b - a\) if \(b - a \lt 0\).
Solution
Recall that if \(x > 0\), then
\[ |x| = x\]
If \(b - a \lt 0\), then \(a - b > 0\) and
\[ |a - b| = a - b\]
The statement is FALSE.
-
(True or False) \(|x^2 + 1| = x^2 + 1\).
Solution
Since \(x^2 + 1\) is positive for all real \(x\), then
\[ |x^2 + 1| = x^2 + 1\]
The statement is TRUE.
-
(True or False) \(\sqrt{(x - 5)^2} = x - 5\).
Solution
Let \[x = -4\]
Left Hand Side: \(\sqrt{(-4 - 5)^2} = \sqrt{81} = 9\)
Right Hand Side : \(-4 - 5 = -9\)
The statement is FALSE.
-
(True or False) \((x - 2)(x + 2) = x^2 - 4x - 4\).
Solution
Expand and simplify
\[ (x - 2)(x + 2) = x^2 + 2x -2x - 4 = x^2 - 4\]
The statement is FALSE.
-
(True or False) \(\sqrt{x + 9} = \sqrt{x} + \sqrt{9}\), for all \(x\) real.
Solution
Let \[ x = 16\]:
Left Hand Side: \(\sqrt{25} = 5\),
Right Hand Side: \(\sqrt{16} + \sqrt{9} = 4 + 3 = 7\)
The statement is FALSE.
-
(True or False) \(|x - 3| = |x| + |3|\), for all \(x\) real and negative.
Solution
Since \(x \lt 0 \) , then \( 3 − x >0 \) and hence:
\[ |x - 3| = |-(3 - x) | = 3 - x = |3| + |x|\]
The statement is TRUE.
-
(True or False) \((x + 2)^3 = x^3 + 2^3\), for all \(x\) real.
Solution
\[ (x + 2)^3 = x^3 + 6x^2 + 12x + 8\]
The statement is FALSE.
-
(True or False) If \(k = 4\), then the equation \(x^2 - kx = -4\) has one solution only.
Solution
\[x^2 - 4x + 4 = 0 \Rightarrow (x - 2)^2 = 0\]
One solution: \(x = 2\)
-
(True or False) The discriminant of the equation: \(2x^2 - 4x + 9 = 0\) is negative.
Solution
The dicriminat \( \Delta \) is given by:
\[
\Delta = (-4)^2 - 4(2)(9) = 16 - 72 = -56
\]
The discriminant is negative. The statement is True.
-
(True or False) The degree of \(P(x) = (x - 2)(-x + 3)(x - 4)\) is equal to -3.
Solution
\[
P(x) = (-x^2 + 5x - 6)(x - 4) = -x^3 + 9x^2 - 26x + 24
\]
Degree is 3. The statement is FALSE.
-
(True or False) The distance between the points \((0, 0)\) and \((1, 1)\) is equal to 1.
Solution
\[
\text{Distance} = \sqrt{(1 - 0)^2 + (1 - 0)^2} = \sqrt{2}
\]
-
(True or False) The slope of the line \(2x + 3y = -2\) is negative.
Solution
\[ 3y = -2x - 2 \Rightarrow y = -\frac{2}{3}x - \frac{2}{3}\]
Slope is \(-\frac{2}{3}\)
-
(True or False) The relation \(2y + x^2 = 2\) represents \(y\) as a function of \(x\).
Solution
\[ y = -\frac{x^2}{2} + 1\]
One output per input → function. Statement is TRUE.
-
(True or False) The relation \(2y + x^2 = 2\) represents \(x\) as a function of \(y\).
Solution
\[ x^2 = 2 - 2y \Rightarrow x = \pm\sqrt{2 - 2y}\]
Two values for one \(y\) → NOT a function.
-
(True or False) The relation \(|x| = |y|\) does NOT represent \(x\) as a function of \(y\).
Solution
Solve \(|x| = |y|\) for \( x \):
\[ x = \pm y\]
Two values of \(x\) per \(y\) → NOT a function.
-
(True or False) The relation \(|x| = |y|\) does NOT represent \(y\) as a function of \(x\).
Solution
Solve \(|x| = |y|\) for \(y\): \[y = \pm x\]
Two values of \(y\) per \(x\) → NOT a function.
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