Answers to Sample Exam Problems
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1 Math Answers to Sample Exam Problems () Find the absolute value, reciprocal, opposite of a if a = 9; a = ; Absolute value: 9 = 9; = ; Reciprocal: 9 ; ; Opposite: 9; () Commutative law; Associative law; (c) Distributive law; () Find the slope of the line passing through (, ) and (, ); whose equation is x + y = ; By Slope Formula: = m = y y x x = ( ) = Write the equation to Slope-intercept form: = y = x + = y = x + = the slope of this line is: m = () Graph in the rectangular system: a line with slope m = and y-intercept b = ; the equation x = y ; (c) the equation y = x ; Start from the y-intercept (0, ), since the slope is, going down units and going right units to reach the point (, ); then, draw a line passing through the two points: (0, ) and (, ); Rewrite the equation to Slope-intercept form: x = y = y = x + = y = x + = m = and b = ; then, following thy way in to draw the line; (c) The equation is in Slope-intercept form: y = x = m = follow the way in to draw the line; () What is the equation of the line: if it passes through (, ) and (, 9)? if it is with m = and b = ; (m is the slope, and b is the y-intercept.) Find the slope first: = m = y y x x = 9 ( ) = 0 = ; and b =, then, then, use the slope and the point (, ) to write the equation = y ( ) = (x ) = y + = (x ) (You can also use the slope and the other point (, 9) to write the equation. Apply the Slope-Intercept form: = y = ( )x + ( ) = y = x () What is the degree of the monomial 7x y; the polynomial x y x y + 9x y? The degree of a monomial is the (sum of) the exponent(s) of the variable(s); = Its degree is: + = The degree of a polynomial is the highest degree among the monomials contained in the polynomial: = Its degree is 7 (7) Graph on the number line x < ; > x ; x < 0 > x ; () Evaluate x x+ x Or 0 x x+ x = ( 0.) ( 0.)+ ( 0.).7 = 7 = x y+7 y x for x = 0. for x = and y = = (0.) ( )+ (0.) = =.7.7. x y+7 y x = ( ) ( )+7 ( ) ( ) = = =
2 x (9) What x value(s) will make x x undefined? Find the numbers that make the denominator zero: Set x x = 0 = (x )(x + ) = 0 = x = and x = will make x x x undefined (because its denominator will be zero if x = or x =.) (0) Factor a 7ab ; 0x + x x ; (c) x xy ; a 7ab = a(a 9b ) = a(a + b)(a b); 0x + x x = x (x + 9x ) = x (x )(x + ); (c) x xy = x(x 7y ) = x((x) (y) ) = x(x y)(x + xy + 9y ) () Simplifying: 0x y 7 z 0x y 7 z = ( )()x y 7 z = ( )()()(x )(y )(y)(z 0 )(z) = x y z 0yz () Simplifying: 7x + x x 7x 7x + x x 7x = (x x) + (x x) x( x) = x x + x x x x = x x () Simplifying: x x (x ) 7 ( ) x x (x ) = x x x ( )( ) = x x x = x +( )+ = x 0 = (c) 7 ( ) = +( 7) = = = ( ) = 0 ( a a a 7 a ) = ( a a ) = (a ) = a = a () Simplifying 9 ; ; (c) ; (d) ( 9 ) 9 = ( 9) = () = 7 = (c) = = = (d) ( 9 ) = ( 9 ) = 9 = 9 7 = ( = = ) () () Simplifying ( ) (a b)(a + b) (a b) (c) (x x + )(x x ) (d) ( x y) ( ) = ( ) ( )() + () = ( ) 0 + = 0 + = 7 0 (a b)(a+b) (a b) = (a (b) ) ((a) (a)+ ) = (a 9b ) (9a ab+b ) = a 9b 9a + ab b = a + ab 0b (c) (x x + )(x x ) = x 7x + 9x x 0 (d) ( x y) = ( x) ( x)( y) + ( y) = x xy + y () Solve the inequality and graph the solution set on the number line: 7 x > (x + ) 7 x > (x + ) = 7 x > x = x + x > 7 = x > 9 (divide both sides by ( ), remember to reverse the inequality symbol) = x < (7) Rationalizing the denominator: 0 = 0 = 0 = + = () Adding: + = ( ) x x + x 0 ; + ( = ( ) = ( ) = ( ) ) 9 subtracting: y+ y+ y +y x + = x + ( ) = x + = x x x x ( x)( ) x x x (c) Dividing: = (x+)(x ) x x 9 x +x x 7x+0 x 0 = x +
3 (c) y+ = y+ = (y+) y = y +y y+ y +y (y+) y(y+) (y+) y y(y+) y(y+) x 9 x +x x 7x+0 x 0 = (x+)(x ) (x )(x+) (x )(x ) = (x+)(x ) (x+)(x ) (x )(x+) = (y+)(y ) y(y+) (x+)(x ) (x )(x ) = y y = (x+) x (9) Simplifying: 7 ( ( )); ( ) 7 ( ( )) = 7 ( ( 9)) = 7 ( +7) = 7 () = 7 00 = 07 ( ) = 9 = (0) Simplifying the complex fraction: Or: 9 = = ; x + x x ; = ( ) ( ) = ( 9 ) ( ) = 7 = 7 = = ( )() 9 = = ( )() 0 x + x = ( x + x )x x ( x )x = x x+ x () Simplifying x ( (x )); = (x )(x ) = x (x+)(x ) x+ x ( (x )) = x ( x + ) = x (7 x) = x + x = 9x () Graph the inequality x y in the rectangular system; Step : Draw the line with the equation x y =, (draw a solid line, do you know why?) Step : Pick a test point, say, (0, 0). It satisfies the inequality x y < (0 0 < is true) Step : Shade the half plane that contains the test point (0, 0), the half plane is the solution set. y 0 x () Solve the equation: x = 7 ( x) x = 7 ( x) = x = x = x = 7 + x = x x = 7 + = x = = x = () Solve the equation: x + x = x x = 0 (by Factoring) x + x = = x + x = 0 = (x )(x + ) = 0 = x =, x = ; (by the Quadratic Formula) = x = b± b ac a x + x = = x + x = 0 = a =, b =, c = = ± ()( ) = ± + = ± 9 = ± () = x = + = =, x = = = x x = 0 = a =, b =, c = = x = b± b ac a () Solve the equation: (x + 7) = = ± ( ) ()( ) = ± +0 = ± 7 = ± 9 = ± 9 () (by Square Root Property) (x + 7) = = x + 7 = ± = x + 7 = ± = x = 7 ± = x = 7±
4 () Solve the equation: x = x (Isolate the radical:) = x = x + (then, square both sides:) = ( x) = (x + ) = x = x + x + 9 = 0 = x + x x = x + 9x + = 0 = (x + )(x + ) = 0 = x =, x = (Check the solutions in the original equation:) Check x = = ( ) = ( ) + = = = = (True); Check x = = ( ) = ( ) + = = = = (False); = x = is solution. (x = is NOT a solution, drop it.) (7) Solve the equation: x + x = x 7x+ + x x = x x x + x = (multiply both sides by LCD x(x ) to eliminate the denominators:) = x + (x ) = x(x ) = x + x = x x = 0 = x x x x + = 0 = x 0x + (switch left and right hand side and divide both sides by to simplify) = x x + = 0 = (x )(x ) = 0 = x =, x = (Both solutions check, see below.) Check x = = + = = + = = = (True) Check x = = + = = + = = = (True) x 7x+ + x x = x x Since the st denominator is: x 7x + = (x )(x ), the LCD is (x )(x ); Multiply both sides by LCD (x )(x ) to eliminate the denominators: = + x(x ) = x(x ) = + x x = x 9x = + x x x + 9x = 0 = x + x + = 0 Multiply both sides by (-) to make the leading coefficient positive: = x x = 0 = (x )(x + ) = 0 = x =, x = The solution is x = (x = is NOT a solution, can you see why?) () Solve the equation for y: Ax + By + Cz = D (y is the variable, everything else is regarded as a constant.) Isolate the variable term: = By = D Ax Cz Divide both sides by B: = y = D Ax Cz B (9) Solve the system of equations: x y = E x y = E E () and, E ( ) = x 0y = 0 E x + 0y = E E + E = 9x = = x = Substitute x = in E = () y = = 0 y = = y = = The solution of the system is: (, ) (0) If working alone, A can complete a task in hours and B can complete the same task in 9 hours. How many hours do they need to complete the task if they work combinedly? Working alone, in EACH hour, A can complete portion of the task; while B can complete portion of the task. 9 Working together, if x hours are needed. Then, in EACH hour, they together can complete portion of the task. x the equation is set up as: + 9 = x To solve it, multiply both sides by LCD (x), = x + x = = x = = x = =. (hours) ( hours and minutes)
5 () An investment of $000 was put in two accounts. At the end of the year, it was found that the st account earned % interest rate and the nd account earned %, and the total interest is $0. Find the original investment amount in each account. Let the amount of investment in the st account be $x, then the amount of investment in the nd account is $(000 x), Account Investment Interest rate Interest st x x nd (000 x) (000 x) According to the problem, the total interest is $0 = 0.0x + 0.0(000 x) = 0 To solve the equation: multiply both sides by 00 = x + (000 x) = 000 = x x = 000 = x = = x = 000 = x = 000 (dollars) The investment in the st account was $000, and the investment in the nd account was ($000 $000 ) = $000. Or, set up two variables to solve the problems: Let x be the amount of investment in the st account and y be the amount of investment in the nd account. x + y = 000 then, set up a system to solve for x and y. 0.0x + 0.0y = 0 () A chemist needs to mix an % salt solution with a % salt solution to obtain liters of a % salt solution. How many liters of each of the salt solutions must be used? Assume that x liters of % solution are needed, then, the amount of % solution is ( x) liters. Solution amount (in liter) percentage of salt salt in the solution % x 0. 0.x % ( x) 0. 0.( x) the mixture =. Since the salt in the mixture is from the two solutions, then, the equation is set as: = 0.x + 0.( x) =. To solve the equation, multiply both sides by 00 first to eliminate the decimals, = x + ( x) = = x + 0 x = = 7x = 0 = 7x = 0 = x = (liters) The chemist used liters of % solution and liters of % solution. Or, set up two variables to solve the problems: Needed: x liters % solution and y liters of % solution, x + y = then, set up a system 0.x + 0.y = 0. to solve for x and y. () Find the length and width of a rectangle if its perimeter is in., and its length is in less than twice its width. Let its length be l and its width be w, l + w = then, set up a system: l = w solve the system = w =, l =
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