# Solutions and Explanations to Intermediate Algebra Questions in Sample 3

Solutions with full explanations of the intermediate algebra questions in sample 3 are presented.

 Write 1.5 � 10-5 in standard form. Solution 1.5 � 10-5 = 1.5 / 10 5 = 1.5 / 100000 = 0.000015 Evaluate: 30 - |-x + 6| for x = 10 Solution Substitute x by 10 in the given expression and evaluate 30 - |- (10) + 6| = 30 - |-10 + 6| = 30 - | - 4 | = 30 - 4 = 26 Evaluate: 2xy3 + x - 2y for x = 2 and y = -2 Solution Substitute x by 2 and y by - 2 in the given expression and evaluate 2(2)(-2)3 + 2 - 2(-2) = 4(-8) + 2 + 4 = - 26 What is the slope of the line perpendicular to the line y = - 4 Solution y = - 4 is a horizontal line and any line perpendicular to it is vertical and therefore has an undefined slope. Write an equation of the line with slope 2 and x-intercept (- 4 , 0). Solution A line with slope m and passes through point (a , b) has an equation of the form. y - b = m(x - a) , point slope form of a line Use the given slope and the point to write the equation of the line. y - 0 = 2(x - (-4)) y = 2x + 8 Solve the equation: -3(-x + 5) + 20 = -10(x - 3) + 4 Solution Expand the expressions with brackets in the given equation 3x - 15 + 20 = -10x + 30 + 4 Group like terms 3x + 5 = -10x + 34 Solve 13 x = 29 , x = 29 / 13 Solve the inequality: 4(x - 6) + 4 < 8(x - 4) Solution Expand the expressions with brackets in the given inequality 4x - 24 + 4 < 8x - 32 Group like terms 4x - 20 < 8x - 32 Solve the inequality 4x - 8x < -32 + 20 -4x < -12 x > 3 Solve the equation: 3(x - 2)2 - 12 = 0 Solution Rewrite equation with square on one side 3(x - 2)2 = 12 (x - 2)2 = 4 x - 2 = (+ or -) 2 x = 2 + 2 = 4 , x = 2 - 2 = 0 The solutions of the given equation are x = 4 and x = 0 Solve the equation: x / 3 + 2 / 7 = x / 7 - 5 Solution The given equation has denominators that need to be cleared by multiplying all terms of the equation by the lowest common denominator of the denominators 3 and 7 which is 21 21(x / 3 + 2 / 7) = 21(x / 7 - 5) Expand and simplify 7x + 6 = 3x - 105 Solve for x 4x = - 111 , x = - 111/4 Line L is defined as line through the point (2 , 7) and perpendicular to the line x + y = 0. What is the point of intersection of L and the line x + y = 0? Solution We first find the slope of the line x + y = 0. Rewrite in slope intercept form y = - x , slope = - 1 Line L is perpendicular to line x + y = 0 and therefore the product of its slope m and the slope of the line x + y = 0 is equal to -1. Hence m*(-1) = -1 , solve for m: m = 1 Let A(a , b) be the point of intersection. Point A and the given point (2 , 7) lie on line L and therefore the slope calculated using point (a , b) and (2 , 7) must be equal to the slope of line L which is - 1. Hence (7 - b) / (2 - a) = 1 Point A(a , b) lie also on line x + y = 0. Hence a + b = 0 Solve the two system of equations (7 - b) / (2 - a) = 1 and a + b = 0 to find point A(a , b). a + b = 0 implies that a = - b Cross multiply (7 - b) / (2 - a) = 1 to obtain 7 - b = 2 - a Substitute a by - b (a = - b above) in the above equation and solve 7 - b = 2 - (-b) , b = 5 / 2 Use a = - b (see above) to find a a = - 5/2 The point of intersection is A(- 5/2 , 5/2) What is the point of intersection of the lines: x + 2y = 4 and -x - 3y = -7? Solution Since the point of intersection lie on the two lines, its coordinates x and y satisfy the two equations simultaneously and are therefore found by solving the system of equations of the two lines. Let us add the right hand side and left hand side of the two equations x + 2y = 4 + - x - 3y = -7 - y = - 3 , equation obtained y = 3 Substitute y by 3 in one of the equations to find x. x + 2(3) = 4 , solve for x: x = - 2 The point of intersection is given by its coordinate as follows (-2 , 3) How many solutions do the system of equations 2x - 3y = 4 and 4x - 6y = -7 have? Solution Multiply all terms of the equation 2x - 3y = 4 to obtain 4 x - 6 y = 8 Subtract right sides and left sides the equation obtained above from the equation 4x - 6y = -7 (4 x - 6 y) - (4x - 6y) = 8 - (-7) Simplify 0 = 15 The above statement is false and therefore the system of equations has no solution. For what value(s) of A does the system of equation A x + 6y = 0 and 2x - 7y = 3 have no solutions? Solution Solve equation A x + 6y = 0 for y y = - A x / 6 Substitute y by - A x / 6x into the equation 2x - 7y = 3 2 x - 7(- A x / 6) = 3 Solve the above for x x = 3 / (2 + 7 A / 6) x is a not a solution if the denominator 2 + 7 A / 6 is equal to zero 2 + 7 A / 6 = 0 Solve for A A = - 12 / 7 Solve |2x - 4| - 2 = 6. How many solutions does the equation 2x2 + 3x = 8 have? Solve the equation 3x2 + 6x - 1 = 8. Solve the system of equations: 2x + 5y = 18 and -3x - y = -1. What is the range of function f defined by: f = {(2,3),(1,4),(5,4),(0,3)} Factor the expression 2x2 + 3x + 1. Factor the expression 10x2 + 20x - 80. Answers to the Above Questions 0.000015 26 -26 undefined (vertical line) y = 2x + 8 29/13 x > 3 {0 , 4} -111/4 (-5/2 , 5/2) (-2 , 3) no solutions -12/7 {-2 , 6} 2 real solutions {-3 , 1} (-1 , 4) {3 , 4} (2x + 1)(x + 1) 10(x - 2)(x + 4) Algebra Questions and problems More ACT, SAT and Compass practice