Answer :
8) Absolute value is the distance from zero on a number line (and has no reference to which direction left/right from zero), so this means the value is always positive:
abs(2) = 2
abs(-2) = 2
9) Solve the equation for x
(3x + y)/z = 2
*multiply both sides by z
(3x + y) = 2z
*subtract y from both sides
3x = 2z - y
*divide both sides by 3
x = (2z - y)/3
10) Which point is a solution to the equation 6x - 5y = 4? Justify your choice
A. (1, 2)
B. (1, -2)
C. (-1, -2)
D. (-1, 2)
*plug (x, y) coordinates into equation and see if the result is a valid equation:
*start with A. (1, 2):
6(1) - 5(2) = 4
6 - 10 = 4
-4 = 4 [NO GOOD]
*now try B. (1, -2):
6(1) - 5(-2) = 4
6 - (-10) = 4
6 + 10 = 4
16 = 4 [NO GOOD]
*now try C. (-1, -2):
6(-1) - 5(-2) = 4
-6 - (-10) = 4
-6 + 10 = 4
4 = 4 [OK]
*just for fun let's also verify D. (-1, 2) is not the solution, since we found that C. was:
6(-1) - 5(2) = 4
-6 - 10 = 4
-16 = 4 [NO GOOD]
The answer is C. (-1, -2) (and the justification is that we solved for it to be true)
11) Domain is all values 'x' (i.e. input)
Range is all values 'y' (i.e. output)
a.) y = 2x + 1 is a line with a slope of 2:1 (vert:horiz) and a y-intercept of y = 1, but because it is a line, it extends from -infinty to +infinity for both 'x' and 'y', so . .
Domain = (-infinity ≤ x ≤ +infinity)
Range = (-infinity ≤ y ≤ +infinity)
b.) This table on shows discrete values of input/output, so the domain/range is also discrete . .
Domain = (3, 7, 11)
Range = (-1, -3, -5)
c.) Just from visual confirmation of the plot's extents . .
Domain = (-5 ≤ x ≤ 5)
Range = (-1 ≤ y ≤ 1)
d.) Again using visual confirmation of the plot's extents . .
Domain = (-2 ≤ x ≤ 2) *note extents are limited by vertical asymptote
Range = (-infinity ≤ y ≤ +infinity)
12) There are 2 lines of symmetry (they are the vertical line drawn at x = 0, and the horizontal line drawn at y = 0 that bisect the ellipse)
abs(2) = 2
abs(-2) = 2
9) Solve the equation for x
(3x + y)/z = 2
*multiply both sides by z
(3x + y) = 2z
*subtract y from both sides
3x = 2z - y
*divide both sides by 3
x = (2z - y)/3
10) Which point is a solution to the equation 6x - 5y = 4? Justify your choice
A. (1, 2)
B. (1, -2)
C. (-1, -2)
D. (-1, 2)
*plug (x, y) coordinates into equation and see if the result is a valid equation:
*start with A. (1, 2):
6(1) - 5(2) = 4
6 - 10 = 4
-4 = 4 [NO GOOD]
*now try B. (1, -2):
6(1) - 5(-2) = 4
6 - (-10) = 4
6 + 10 = 4
16 = 4 [NO GOOD]
*now try C. (-1, -2):
6(-1) - 5(-2) = 4
-6 - (-10) = 4
-6 + 10 = 4
4 = 4 [OK]
*just for fun let's also verify D. (-1, 2) is not the solution, since we found that C. was:
6(-1) - 5(2) = 4
-6 - 10 = 4
-16 = 4 [NO GOOD]
The answer is C. (-1, -2) (and the justification is that we solved for it to be true)
11) Domain is all values 'x' (i.e. input)
Range is all values 'y' (i.e. output)
a.) y = 2x + 1 is a line with a slope of 2:1 (vert:horiz) and a y-intercept of y = 1, but because it is a line, it extends from -infinty to +infinity for both 'x' and 'y', so . .
Domain = (-infinity ≤ x ≤ +infinity)
Range = (-infinity ≤ y ≤ +infinity)
b.) This table on shows discrete values of input/output, so the domain/range is also discrete . .
Domain = (3, 7, 11)
Range = (-1, -3, -5)
c.) Just from visual confirmation of the plot's extents . .
Domain = (-5 ≤ x ≤ 5)
Range = (-1 ≤ y ≤ 1)
d.) Again using visual confirmation of the plot's extents . .
Domain = (-2 ≤ x ≤ 2) *note extents are limited by vertical asymptote
Range = (-infinity ≤ y ≤ +infinity)
12) There are 2 lines of symmetry (they are the vertical line drawn at x = 0, and the horizontal line drawn at y = 0 that bisect the ellipse)
