# Solutions and Explanations to Intermediate Algebra Questions in Sample 4

Full explanations of solutions for intermediate algebra questions in sample 4 are presented.

 (True or False)   x 2 and 2 x are like terms. Solution The statement "x 2 and 2 x 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)   1 / (x - 9) = 0 for x = 9. Solution Substitute x by 9 in the expression 1 / (x - 9). 1 / (x - 9) = 1 / (9 - 9) = 1 / 0 = undefined The statement "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 "The set of ordered pairs {(0,0),(2,0),(3,0),(10,0)} represents a function" is TRUE. (True or False)   |a - b| = b - a if b - a < 0. Solution Recall that if x > 0, then | x | = x But if b - a < 0, then a - b > 0 and |a - b| = a - b The statement "|a - b| = b - a if b - a < 0" is FALSE. (True or False)   |x2 + 1| = x2 + 1. Solution Recall that if x > 0, then | x | = x Since x2 + 1 is positive for all values of real x, then | x2 + 1 | = x2 + 1 The statement |x2 + 1| = x2 + 1" is TRUE. (True or False)   √(x - 5) 2 = x - 5. Solution For the above statement to be true, it must be true for all values of x for which the expression are defined. Let x = - 4 and evaluate the right and left hand expressions. Left: √(x - 5) 2 = √(-4 - 5) 2 = √81 = 9 Right: x - 5 = - 4 - 5 = - 9 The above statement is not true for x = - 4 and therefore the statement "√(x - 5) 2 = x - 5" is FALSE. (True or False)   (x - 2)(x + 2) = x2 - 4x - 4. Solution Expand (x - 2)(x + 2) (x - 2)(x + 2) = x2 + 2x - 2x - 4 Group like terms (x - 2)(x + 2) = x2 - 4 The statement "(x - 2)(x + 2) = x2 - 4x - 4" is FALSE. (True or False)   √(x + 9) = √x + √9, for all x real. Solution For the above statement to be true, it must be true for all values of x for which the expression are defined. Let x = 16 and evaluate the right and left hand expressions. Left: √(x + 9) = √(16 + 9) = √(25) = 5 Right: √x + √9 = √(16) + √9 = 4 + 3 = 7 The above statement is not true for x = 16 and therefore the statement "√(x + 9) = √(x) + √(9), for all x real" is FALSE. (True or False)   |x - 3| = |x| + |3|, for all x real and negative. Solution Start with |x - 3| |x - 3| = |3 - x| = |3 + (- x)| The absolute value of the sum of two positive numbers is equal to the sum of the numbers. Since x is negative - x is positive and hence |3 + (- x)| = |3| + |-x| = |3| + |x| The statement "|x - 3| = |x| + |3|, for all x real and negative" is TRUE. (True or False)   (x + 2)3 = x3 + 23, for all x real. Solution Expand (x + 2)3 (x + 2)3 = (x + 2)(x + 2)2 = (x + 2)(x2 + 4 x + 4) = x3 + 6 x2 + 12 x + 8 (True or False)   If k = 4, then the equation x2 - k x = - 4 has one solution only. Solution Set k = 4 in the given equation x2 - 4 x = -4 Solve it x2 - 4 x + 4 = 0 (x - 2)2 = 0 one solution: x = 2 (True or False)   The discriminant of the equation: 2x2 - 4x + 9 = 0 is negative. Solution Calculate discriminant Δ Δ = (-4)2 - 4(2)(9) = 16 - 72 = -56 The discriminant Δ is negative (True or False)   The degree of the polynomial P(x) = (x - 2)(-x + 3)(x - 4) is equal to -3. Solution Note that the degree of a polynomial is never negative. Expand the given polynomial P(x) = (x - 2)(-x + 3)(x - 4) = (-x2 + 5 x - 6)(x - 4) = - x3 + 9 x2 - 26 x + 24 The degree of the given polynomial is the highest power and is therefore 3 (True or False)   The distance between the points (0 , 0) and (1 , 1) in a rectangular system of axes is equal to 1. Solution Distance between the given points is given by √ ((1 - 0)2 + (1 - 0)2) = √2 (True or False)   The slope of the line 2x + 3y = -2 is negative. Solution Write the given equation in slope intercept form y = m x + b and identify the slope m. 3 y = - 2x - 2 y = (-2/3) x - 2/3 slope is equal to -2/3. (True or False)   The relation 2y + x2 = 2 represents y as a function of x. Solution Solve the given equation for y y = - x2 / 2 + 1 For each value of x we obtain one value of y only and therefore y is a function of x. (True or False)   The relation 2y + x2 = 2 represents x as a function of y. Solution Solve the given equation for x x2 = 2 - 2 y x = ~+mn~√(2 - 2y) For any value of y such that 2 - 2y ≥0 we have two values of x and therefore x is not a function of y. (True or False)   The relation |x| = |y| DOES NOT represent x as a function of y. Solution Solve the given equation for x x = ~+mn~y For any value of y; we have two values of x and therefore x is not a function of y. (True or False)   The relation |x| = |y| DOES NOT represent y as a function of x. Solve the given equation for y y = ~+mn~x For any value of x; we have two values of y and therefore y is not a function of x.