Chapter 2 Equations and Inequalities. 2. z = -6 then z + 11 = z + 11 = 5 3. m = -5 then 6 - m = 6 -(-5) 6 - m = 11

Transcription

1 Chapter Equations and Inequalities Section. Definitions and Solving Equations Practice y = (8) - - = - 8 is a solution. - n = (n-) - (6) (6 - ) - () - 6 is not a solution. (x - ) = x - (- -) (-) - (-6) = -8 - is a solution. x - ( x + ) = x +. () -( + ) + - ( + ) is not a solution b + = = 6 is a solution Practice... x = then x + 6 = + 6 x + 6 = 8. z = -6 then z + = -6 + z + =. m = - then 6 - m = 6 -(-) 6 - m =. 7 = s then - -s = s = -0 Practice... y - 8 = 7 y = y + 0 = y = Check: - 8 = 7 7 = = v = v = v = v Check:-7 = = -7. y + = y - 8 y + -y = y y y + = -8 y + -= -8 - y = -7 Check: (-7) + = (-7) = = x = x x - x = x - - x - + x = x + = - + x = Check:- + ()= = 8 8 = 8. x - 6 = (x + ) x - 6 = x + 6 x x =x + 6 -x x - 6 = 6 x = 6 +6 x = 7 Check: (7) - 6 = (7 +) 60-6 = (8) = 6. -x = -(x - ) CHAPTER EQUATIONS AND INEQUALITIES

2 -x = -x + -x +x = -x + +x x = Check: -() = -( - ) - = -(8) - = - Practice.. x. = ( x ) = ( ) x = - Check: = =. y = ( y ) = ( ) y - = -0 y - + = -0 + y = - Check: = 0 = - = = x ( 8 ) = ( ) x -0 = x Check: -8 = ( 0) -8 = (-) -8 = -8 Practice... x = x = x = 7 Check: (7) = =. = -m = m - = m Check: = -(-) =. 0 = -(y -) 0 = -8y = -8y - 8 = -8y 8 8 = 8 y 8 - = y Check: 0 = -((-) - ) 0 = - (- - ) 0 = -(-) 0 = 0. (y - ) + = - y - + = - y - = - y - + = - + y = -8 y = 8 y = - Check:(--) + = - (-) + = = - - = -. 6( - a) + a = (6 - ) - 6a + a = () - a = - a - = - -a = - a = a = Check:6( - ) + () = (6 - ) 6(-) + = () -0 + = = Exercise Set.. - w = - (-) = - is a solution. 0x + (00 - x) = 0 CHAPTER EQUATIONS AND INEQUALITIES

3 0(00) + (00-00) (-00) is not a solution. ( - x) -= x - ( - 7 ) 7 ( 0 7 ) 6 ( ) = 7 is a solution 7. -(6 -x) =x - (x + ) -(6 - ) () - ( + ) -() 0 - (6) is not a solution.. m + ( m ) = ( 0) + ( ( 0) ) 0 + ( ) -0 + = -0 is a solution x + ( x ) = ( ) + ( ) 7 + ( 0) is not a solution. x = x ( ) = - / is a solution. d = - then d + 6 = d + 6 = - 7. z = -. then z +. = z +. =.. z = z + = + z + = then z + = +. y + = y - y + - y = y - - y y + = - y + - = - - y = x = - 8x - x + x = - 8x + x = + x - = + x - = x. 7. x 6 = 6 ( x 6 ) = ( 6 ) x = - w 0 = CHAPTER EQUATIONS AND INEQUALITIES

4 ... 0( w 0 ) = 0( ) w = s = 6 ( ) s = ( 6 s = - x = ( x ) = ( ) x - = -0 x - + = -0 + x = -7 8 a = 8 ( 8 ) a = 8 ( ) a = 8 a =. - = t -(-) = -( t ) 6 = t 7. - = 7x 7 = 7x 7 - = x. -6 = -8x 6 8 = 8w 8 = w. x + = 7 x + - = 7 - x = x = x = 7 ) 8 = x - = x. y + = y + y + - y = y + - y y + = y + - = - y = 0 y = 0 7. x - = x - x - - x = x - -x x - = - x - + = - + x = x = x = 7. -( - n) = -n -0 + n = -n -0 + n - n = -n - n -0 = -7n 0 7 = 7 n 7 0 =n 7. -(y - 7) = -0y -y + = -0y -y + +y = -0y + y = y = y 7 = y. x. 8 m. 6. -y is a product whose factors are - and y. = 7 - x - 7 = 7 - x = -x = = 0 CHAPTER EQUATIONS AND INEQUALITIES

5 8. The Distributive Property of Multiplication over Addition. y + y = y( + ) 60. -(x -x) = -(-x) = x = = = = 0 6. ( - ) = () = () = 6. x + y - x = -x + y Section. More on Solving Equations Practice... = (x + ) - x - 0 = x x - 0 = x = x = x. -( - x) + = x x + = x x = x- - + x - x = x - - x - + x = x + = - + x = 6 x = 6 x =. a - (a - ) + = - ( - a) a - a + + = a -a + = - + a -a + + a = - + a + a = - + a + = - + a + = a = a = a. 7(y + ) + y = - + y - (y + ) 7y + + y = - + y - y - y + = -6 + y y + - y =-6 + y - y 0y + = -6 0y + - = -6-0y = -0 0y 0 = 0 0 y = -. - ( - m) = (m + ) - (m - ) m = m + - m m = m m - m = m + - m - + m = - + m + = + m = 6 m = 6 m = Practice.. x. 7 0 = 0 LCD = 60 60( x ) 60( 7 0 ) = 60( 0 ) x - (7) = 6 x - = 6 x - + = 6 + x = 0 x = 0 x =. + n = 0 LCD = 0 0( ) + 0( + (n) = - + 6n = - + 6n - = - - 6n = -6 6n 6 = 6 6 n = - n ) = 0( 0 ) CHAPTER EQUATIONS AND INEQUALITIES

6 ... x + = + x LCD = ( x ) + ( ) = ( ) + ( x ) x + = - + x x + - x = - + x - x x + = - x + - = - - x = -0 x = 0 x = - b + = b LCD = ( b b ) + ( ) = ( ) ( ) b + = b - 8 b + - b = b b b + = -8 b + - = -8 - b = -60 b = 60 b = -0 ( + x ) + ( x ) = LCD = ( ( + x )) + ( ( x )) = ( ) ( + x) + ( - x) = + 6x x = x + = x + - = - x = - Practice... (x +.0) + 0x = 0 x x = 0 x = 0 x = x = 87.0 x = x =.. 0.0x + 0.(7) = 0.(x + 7) 0.0x +. = 0.x x+.-0.0x = 0.x x. = 0.0x = 0.0x = 0.0x = 0. 0x 0. 0 = x. 0.06x+0.0(000-x)= x x=80 0.0x +70=80 0.0x+70-70= x= x 0. 0 = x = x+0.(000-x)= x+0-0.x = x+0= x+0-0= x= x 0. 0 = x = 0 Practice... + (x - ) = x x + x - = x x = x x - x = x x -7 = -7 This is an identity and the solution is all real numbers.. - (y - ) = y + - 8y - y + 6 = -y + - y = -y + - y + y = -y + + y = This is a contradiction and there is no solution (x - ) = 6x x 7 + x - = x x = x x - x = x x = 8 This is a contradiction and there is 6 CHAPTER EQUATIONS AND INEQUALITIES

7 no solution.. 8x - = x - (x + ) + 8x - = x - x - + 8x - = -x - 8x - + x = -x - + x 0x - = - 0x - + = - + 0x = 0 0x 0 = 0 0 x = (x + ) = x - - 6x 6 - x - 0 = -x x = -x x + x = -x - + x - = - This is an identity and the solution is all real numbers. 6. 7x = -x 7x + x = -x + x 8x = 0 8 x 8 = 0 8 x = 0 Exercise Set.. + x = 6( x+ 7) - + x = 6x x = 6x + x - x = 6x - x = x = x 8 = x. + x = -(x + 8) + + x = -x x = -x + x - x = -x - x = -x = x = x. -(x + ) + x = -x + 7 -x -8 + x = -x + 7 no -x -8 = -x + 7 -x -8 +x = -x x -8 = 7 This is a contradiction and there is solution x = -( x+ ) x = -x x = -x -6 + x - x = -x - x -6 = -x 6 = x 6 x = = (y - ) + 6(y - ) = -y + + 6y - = y = y = y =. 8(a - ) + ( - a) + 0 = - 8a a + 0 = - a = - a = a = -. 6 = -(y - (6y-6)) 6 = -(y - 6y + 6) 6 = -(-y + 6) 6 = 0y = 0y = 0y 0 0 = 0y 0 = y. -(a - (a - )) = - -(a - a + ) = - -(-a + ) = - a - = - a - + = - + a = a = a = = CHAPTER EQUATIONS AND INEQUALITIES 7

8 (x - ) = x - (x + ) -7 - x + = x - x x = -x x + x = -x + x = x = x = x. (m - 6) - 6 = -7 + (m - ) m - -6 = -7 + m - 6 m - 8 = - + m m m = - + m - m -8 = - + 7m -8 + = - + 7m + = 7m 7 = 7 m 7 7 = m. -6(k -) - = - + ( -k) -6k =- + - k -6k + 6 = - k -6k k = - k + 6k 6 = - 6k 6 - = - 6k - -7 = -6k 7 6 = 6 k = k. ( + (s - )) = (s - ) - 6s ( + s - 6) = s - - 6s (- + s) = -s s = -s s + s = -s- + s - + 8s = s + = - + 8s = - 8 s 8 = 8 s = 8 =. -n + 8(6 - n) = + n - ( + n) -n + 8-8n = + n - - n -n + 8 = -n = - 8 -n = -6 n = 6 n = 6 = 7. a + 7(-8-a) = - a - ( - a) a - 6-7a = - a - + a 8a - 6 = -8 8a = a = 8 8 a 8 = 8 8 a =6... x = LCD = ( x ) ( ) = ( ) x - () = - x - 8 = - x = x = - x = x = - + n = 0 LCD = 0 0( ) + 0( + (n) = - + 6n = - + 6n - = - - 6n = -6 6n 6 = 6 6 n = - n ) = 0( 0 ) b + = b LCD = ( b b ) + ( ) = ( ) ( ) b + = b - 8 b + - b = b -8 - b b + = -8 b + - = -8 - b = CHAPTER EQUATIONS AND INEQUALITIES

9 . b = 60 b = -0 x + ( x ) = LCD = ( x ) + ( ( x )) = ( ) (x) + (( - x)) = 60 x + (6 - x) = 60 x + 0-0x = 60 x + 0 = 60 x = 60-0 x = 0 x = 0 x =. 7a - 6 = - + 6a 7a - 6-6a = - + 6a - 6a a - 6 = - a = a = - ( x ) = ( x + ) 0( ( x )) = 0( ( x + )) 0( ) (x - ) = (x + ) - (x - ) = x + - x - 0 = x - 8 x - x - 0 = x - x - 8 x - 0 = -8 x = x = x = x = 7. 6 r + ( r) = ( 6 r ) + ( ( r) ) = ( ) (r) + ( - r) = 0r + ( - r) = 0r r = r + 6 = r = - 6 r = -. ( x ) = x +. ( x ) ( ) = ( x ) + ( ) 8(x - ) - = x + 8x = x + 8x - 6 = x + 8x x = x + - x x - 6 = x = +6 x = 80 x = 80 x = 80 7 a = + a ( 7 a ) ( ) = ( ) + ( a ). 7. x x + 6 ( x = 6 ( x + ) ) 6 ( x + (x -) -(x + ) = x + 6x - - x - = x + x - = x + x - - x = x + - x x - = x - + = + x = x = x = ) = 6 ( 6 ( x + ) ) ( 0 x ) ( x ) = ( x ) + x 0 ( 0 x ) 0 ( x ) = 0 ( x ) + 0 x (0-x)-0(-x)=(x-)+x 0-0x x = x x 0x = 0x - 0 0x - 0x = 0x - 0-0x 0x = -0 0x 0 = 0 0 x = - CHAPTER EQUATIONS AND INEQUALITIES

10 . (x -.7 ) = x x -.7 = x x -.7 -x= x - x -.7 = 0x = x = 0x x=0.07(x - ) 0.0x =0.07x x -0.07x= x = x 0. 0 = x = x+0.(x - 7) =. 0.x +0.x - 8.=. 0.6x -8. =. 0.6x = x=0 0.6x 0. 6 = x = 0. (n -.7) + 80(n +.) = 0 n n + 00 =0 n + 8.7= 0 n = n = 7. n = 7. n = (x+.0) - 8(x - 0.0) = 8.8 x x = 8.8 -x + 0. = 8.8 -x = x = 8. -x = 8. x = t +0.(6 - t) = 0.0(t - ) 0.0t t = 0.0t t +.0 = 0.0t t+.0-0.t = 0.0t t.0 =0.t =0.t =0.t = 0.t 0. 0 = t 6. x -(7 + x) = -(x + ) - x - - x = -x x - = -x - -x - + x = -x - + x x - = - x - + = - + x = 8 x = 8 x = 6. + ( x + 8 ) = x x + ( x + 8 ) = ( x ) ( x ) 6 + (x + 8) = -x + + x 6 + x + 6 = x + + x = x + + x - x = x + - x = This is an identity. The solution is all real numbers. 6. x = x x - x = x - x x = 0 x 0 = x= y+0.(y +0) = 0.8(y+ 0) 0.y + 0.y + 6. =0.6y + 0.y +6. =0.6y + 0.y y =0.6y +-0.y 6. =0.0y =0.0y + -. =0.0y = 0.0y 0. 0 =y = 6 = < > 0 CHAPTER EQUATIONS AND INEQUALITIES

11 The Commutative Property of Multiplication 7. The base is 6 and the exponent is. 6 = 6 7. x - 7 = - (x - ) () ( - ) () 8-8 = 8 is a solution = = + = 7 = 7 + = = = (x - ) - (x - ) = -x + - x + = -6x = = [ + (/) -]= + 6[ + () - ]= + 6[ + - ]= + 6[ ]= + 6[]= + 0= = 80 = = 0 = = = = = = = 6 = 7 8 ( ) = 7 8 ( ) = 7 = 7 6 = CHAPTER EQUATIONS AND INEQUALITIES

12 Section. Literal Equations and Formulas Practice... This is solved for a in terms of p, r and t.. This is solved for A in terms of b and h.. This is solved for V in terms of r and h. This is solved for y in terms of x. Practice... i = prt i pt = prt pt i pt = r. PV = nrt PV nt = nrt nt PV nt = R. M = a + b ( M ) = ( a + b ) M = a + b M - a = a + b - a M - a= b. a = p + prt a - p = p+ prt - p a - p = prt CHAPTER EQUATIONS AND INEQUALITIES

13 a - p = prt pr pr a - p = t pr. ax + by + c = 0 ax + by + c - ax - c = 0 - ax - c by = -ax - c = -ax - c by b y = b -ax - c b 6. F = C + ( F ) = ( C ) + ( ) F = C + 80 F - 80 = C F - 80 = C F - 80 = C C ( F - ) = C Practice... i = prt i = p( i pt )t i = i. c = πr c = π( c π ) c = c. a = p + prt a = p + pr( a p ) pr a = p + a - p a = a. ax + by + c = 0 ax + b( ax c )+c =0 b ax + -ax - c + c = 0 0 = 0 Exercise Set.. This is solved for I in terms of E and R. This is solved for C in terms of A and B. This is solved for A in terms of t 7. This is solved for d in terms of t and n. This is solved for t in terms of x. A = bh ( A ) = ( bh ) A = bh A = bh b b A = h b. C = 80 - A - B C + A = 80 - A - B + A C + A - C = 80 -B - C A = 80 - B- C. A = 0.h(b+d) A = 0.hb + 0.hd A - 0.hd= 0.hb + 0.hd - 0.hd A - 0.hd = 0.hb A - 0.hd = 0.hb 0. h 0. h A - 0.hd = b 0. h A 0. h d = b 7. V = π r h V = ( π r h ) V = πr h V π r = h CHAPTER EQUATIONS AND INEQUALITIES

14 . A = a + b + c A = ( a + b + c ) A = a + b + c A - a - c = a + b + c - a - c A - a - c = b. h = vt - 6t h + 6t = vt - 6t + 6t h + 6t = vt h + 6t = vt t t h + 6t = v t. H = D N.. H =. ( D N. ).H = DN.H = D N D D.H = N D. s = πrh + πr s - πr = πrh + πr - πr s - πr = πrh s - πr = πrh πr πr s - πr = h πr 7. F = kmm d F d = d ( kmm ) d Fd = kmm Fd km = kmm km Fd km = M. W = LZ P PW = P ( LZ P ) PW = LZ PW W P = LZ W = LZ W. C = 80 - A - B C = 80 - A - (80 - A - C) C = 80 - A A + C C = C. s = 80(n - ) s s = 80( 80 + ) s s = 80( 80 ) s = s. K = mv K = ( K v ) v K=K 7. I = E R I = E E I I = E E I I = E I E I =I. A = ab + bh A = ( A bh ) b + bh b A = A - bh + bh A = A. x / -/ < > CHAPTER EQUATIONS AND INEQUALITIES

15 . -6., -,., 7. 6 = 6 = 6 6 = = = 7 6 = = - + = -6 + = x - [x + (x + ) + 7]= 7x - [x + x + + 7]= 7x - [x + ]= 7x - 0x - 6= -x (x - ) + x = x - - x + + x = x - + x = x - + x - x = x - - x + x = - + x- = - - x = -7. x = + x ( ) ( x ) = ( ) + ( x ) - x = - + x - x+ x = - + x + x = - + x + = - + x + = x = x 6 = x d - 0.(d + ) =. 0.0d - 0.d -. =. -0.0d -. =. -0.0d = d =.7-0.0d 0. 0 = d = -. (x - ) + x = x - x - + x = x - x - = x - x - - x = x - - x - = - This is an identity. The solution is all real numbers.. x = x x - x = x - x x = 0 x = 0. x = ( + x) + (x - ) x = 6 + x + x - x = + x x - x = + x - x 0 = This is a contradiction. There is no solution. Section. Properties and Graphs of Inequalities Practice... x - >0 (-) < 0 - is not a solution. x + < (-) < - is a solution. 6 - y > () -7 > -7 is a solution CHAPTER EQUATIONS AND INEQUALITIES

16 . - m > - () is not a solution.. + x <- + (-8) - -6 < - -8 is a solution. 6. y + > > - - is a solution. Practice... Conditional inequality. Identity. Contradiction. Conditional. Contradiction 6. Identity 6. < < >r 7. < < >p > < >t Practice... p <. n > -. n <. f <. c < $7, m >- Practice... < < >m > < >m. < < >n > < >t. < < >m Practice... y is less than or equal to -6; y is at most -6; the maximum value of y is q is greater than ; q is over ; q has a value that is above.. p is greater than or equal to-8; p is at least -8; the minimum value of p is t is less than ; t is under ; t is below. Exercise Set.. -x -. <. -(-.) CHAPTER EQUATIONS AND INEQUALITIES

17 <. -. is a solution. x + 7> ( ) > / is a solution.. -x < -. + x -(-) -. + (-) > -. - is not a solution. 7. ( -x)< - x ( - ) - (-) - - > - is not a solution.. Identity. Contradiction. Conditional. Conditional 7. < < >n > < >p > < >a. < < >m < >r 7. p < $.0. h < 00.7 feet. w > feet. c < 70 cars. v >-7 7. s > 0%. y is greater than or equal to -.; y is at least -.; the minimum value of y is -.. s is less than -.; s is under -.; s is below -.. y is greater than -.; y is above -.;y is over -.; y is more than -.. s is greater than -.; s is above -.; s is over -.; s is more than b is less than or equal to -; b has a maximum value of -; b is at most -. -,-0.,6.,8 0. Distributive Property of Multiplication over Addition. x + y = 0 x + y - x = 0 - x y = 0 - x y y = = 0 - x 0 - x or y = - x/. 8c - (c - ) + c= 8c - c+ + c = c > CHAPTER EQUATIONS AND INEQUALITIES 7

18 . + ( ) = + ( ) = + = + =. -(x - x) = -x. ( w ) w + w = w w + w = 6 w 6 w + 6 w = 6 w = 7 w = x 60. x - ( - x) = (x - ) + x x = 6x - + 6x - 0 = 6x - 6x - 0-6x = 6x - - 6x -0 = - This is a contradiction. There is no solution. 6. x + [x + 6(x + ) - 8]= x + [x + 6x + 6-8]=x + [8x - ]= x + x - 8 = x x + + ( x ) = x + + x 6 = x + 6 x = 7 x 8. 0.(0 - d) = 0.d + -0.d = 0.d d + 0.d = 0.d + +0.d = 0.8d + - = 0.8d = 0.8d = d -. = d. x + = ( x ) ( x ) + ( ) = ( x ) x + 8 = 6x - x x = 6x - - x 8 = x = x = x 0 = x 8 CHAPTER EQUATIONS AND INEQUALITIES

19 . x - (x - ) x - x + 8 x + 8 x x - Section. Solving Inequalities Practice... x + 8 < - x <- -8 x <-0 <----- < >x x + - x <- + 8 x + < 7 x + - <7 - x < < x - - x x x or x > > 6. -6(8 - ) >-(0 - x) > -0 + x - > -0 + x > -0 + x >x or x< - < Practice... y -6 y -6 y > < >y. 8 > -x 8 < x - < x or x > -. y - (y + ) y - y - y - y y 8 --> < >y > x < ( x ) < ( ) x< - CHAPTER EQUATIONS AND INEQUALITIES

20 . <-- t t -( ) ( ) t - < < >t. -x + >x + -x + + x >x + + x > x + - >x >x - > x - > x or x < - < x < 6 ( x x >-() x > - ) < 6 ( ) > 6. 0 > x ( 0) > -() < x - < x or x > - ( x ) > Practice... x x x + - x 7 + x - x x + 7 x x. m - 6 >m + m m >m + - m -6 > This is a contradiction. There is no solution. < >m. ( - y) y y y -8 -y -8 y > < >y y + >y - 6 y + - y >y y > -6 This is an identity. The solution is all real numbers. < > < >y < (y - ) 6(- -y) y y y y -6 - y + y 0 CHAPTER EQUATIONS AND INEQUALITIES

21 y y y 0 y 0-0x 0 > 0 x < > < >y < (x - ) + < 7( + ) -x < 7() -x + 0< -x < - 0 -x < -x < x > -/ > 8. + x > x x ( ) + ( x ) > ( x ) ( x ) + x >x - x + x >x + x - x >x - x > 0 This is an identity. The solution is all real numbers. < > x 0. x x ( x ) ( x ) ( x ) ( ) x - x 6x - x 6x - x - 6x 6x - - 6x x - x - x - < Exercise Set.. x -.. x x >. x > + 0( ) 0( x ) > 0( ) + 0( ) 6-0x > x > x > 6-0x - 6 > - 6-0x >. y + < ( y ) + ( ) < ( ) y + < y + - < - y <- y < - y < - CHAPTER EQUATIONS AND INEQUALITIES

22 . < < >y y ( y ) ( ) y > < >y 7. -.x >7. -.x. > 7.. x < - < ( - x) - - x - - x x -6 x -6 x < (x - ) + < x - + < x - < x - + < + x < x < x < < x x 0 x + 0( 7 ) 0( 0 ) + 0 x 0x x 0x x 0x + - 0x 7 + 0x - 0x 0x x 0 0x x >. (p - ) + ( - p) > p p > -p + > -p + - > - -p > 0 -p > 0 p < 0 < < >p 7. -(x + 7) - x (x + ) - 0x -x x x + - 0x -x - 7 -x + -x x -x + + x -7 This is an identity. The solution is all real numbers. < >. + 6(x + ) > -x + 6(x + ) + 6x + > -x + 6x + + 6x >x + + 6x - x > x + - x + x > + x - > - x > 0 CHAPTER EQUATIONS AND INEQUALITIES

23 . x > > ( x + ) 6 ( x ) ( )( x + ) ( ) 6 ( x ) (x + /) - (x - /) x + / - 8/ x - / x -/ x - / x -/ - x x -/ - x x -/ -/ x -/ + / -/ +/ x / x < p. + x. x > 7. (c - ) - ( - c) = c c 6c = 8 + = 8 + = + 6 = = (00 -m) + 0.0m = m + 0.0m = m = m - 7 = m = m 0. = m =. x = -x x + x = -x + x x = 0 x = 0. ( x ) ( x ) = ( x 6 ) 6 ( x ) 6 ( x ) = 6 ( x 6 ) 6 8 x - 8-6x + = (x - 6)- 0 x - - 6x + = x x - = x - x - - x = x - - x x - = - x - + = - + x = 0. x + = - (x + ) + 8x x + = - x - + 8x x + = + x x + - x = + x - x = This is an identity. The solution is all real numbers.. x + y = 7 x + y - x = 7 - x y = 7 - x y = 7 - x y = 7 - x or y = 7 x. ( p ) + ( p + ) = p + p + = p 0 + p 0 + = p 0 + CHAPTER EQUATIONS AND INEQUALITIES

24 Chapter Review Exercises. x - (x - ) = -0 Left expression (-) - (- - ) (-) = -0 - is the solution. x = x - = - x - = - The box is -. - = y - y = - (-) - y = + - y = The box is.. 6 = x ( 6 ) = ( x ) 0 = -x 0 = x x = - Right expression. x = 7 ( x ) = ( 7 ) x - = - x - + = - + x = - 6. = w ( ) = ( w ) w = x = -x 8x + x = -x + x x = 0 x = 0 CHAPTER EQUATIONS AND INEQUALITIES

25 8. -(x - ) = - -6x + = - -6x + - = x = -8 6 x 6 = 8 6 x =. - (x - 7) = -(x - ) + - x + = - x x = - x - x + x = - x + x - x = - x - = - -x = -6 x = 6 x = 0. -( - x) = 6(x + ) - x - x + = 6x x 8 - x = x x + x = x x 8 = x = x = x = x.. 0 x = 0 = x = 8 6 ( x + ) ( x ) ( ) = ( 8 ) ( 6 ( x + ) ) (x) - 60 = -6 -(x + ) x - 60 = -6 - x - 8 x - 60 = -0 - x x x = -0 - x + x x - 60 = -0 x = x = - x = x = - x = ( x ) ( ) ( x ) = ( x ) - x = 0(x - /) - x = 0x x + x = 0x + x - 0 = x = x = x = x x = =. x ( x + 8 ) = 6 ( x + ) 6 x 6 ( ( x + 8 )) = 6 ( 6 ( x + )) 6x - (x + 8) = -(x + ) 6x - x - = -x - x - = -x - x - + x = -x - + x 7x - + = - + 7x = 7 7 x 7 = (x+)+ = 0.x 0.x = 0.x 0.x +. =0.x 0.x x =0.x -0.x. =0.x. 0. = 0. x 0. x =. x - (x - ) = - x x - x + 0 = - x -x + 0 = - x -x x = - x + x 0 = This is a contradiction. There is no solution. 6. 8x -(x + ) = 8 -(x + ) 8x - 0x - = 8 - x - 6 -x - = - x -x - + x = - x + x - = This is a contradiction. There is no solution. 7. x + ( x ) = 6 CHAPTER EQUATIONS AND INEQUALITIES

26 x + ( x ) = ( 6 ) x + 6(x - ) = - x + 6x - = - 7x - = - 7x - + = - + 7x = 0 x = x = ( - x) x = - x x = 8 - x 8 + x + x = 8 - x + x 8 + 8x = x - 8 = 8-8 8x = 0 x = 0. This is solved for a in terms of b and c. 0. x + y = 6 x + ( x + ) = 6 x - x + 6 = 6 6 = 6. x - y = x - y + y = + y x = + y x - = + y - x - = y x = y y = x or y = x. I = e + E nr + R ( nr + R ) I = ( nr + R ) ( e + E nr + R ) Inr + RI = e + E Inr + RI -Inr = e + E - Inr RI = e + E - Inr RI I = e + E - Inr I I = e + E - Inr I. a + b = (b + a) a + b = 6b + a a + b - a = 6b + a - a b = -6a b = -6a b = -a. m + (m -p) = 7(p -m) m + m - p = 7p - 7m m - p = 7p - 7m m - p + p = 7p - 7m + p m = 0p - 7m m + 7m = 0p - 7m + 7m m = 0p m 0 = 0p 0 p = m 0 = 6 m. 8 > w < < >w 6. c < < >c 7. - > m < < >m 8. w > < >w. h 7 inches 0. c 8. p > 0. w < pounds. x is larger tjan eight; 8 is less than x; the value of x is over eight. x is greater than or equal to ; x is at least ; the minimum value of x is 6 CHAPTER EQUATIONS AND INEQUALITIES

27 . k is greater than -6; k is more than - 6; k is above ( x+ ) < x x - 6 < x - - x < x - - x + x < x - + x < x - + < x - + <x < x < x or x > 7. x ( x ) + ( x + ) 6 x 6 ( x ) + 6 ( x + ) x 8(x - ) + (x + ) x 8x x + x 0x - x - 0x 0x - - 0x -7x - -7x 7-7 x (x - ) x + ( - x) 8 - x + x + - x 0 - x x x - x x + - x 0 - x 0 - x x -6 -x -6 x Chapter Test. - (x - ) = x ( ) ( ) + 7 -( ) is not a solution. n = -6 + n = n = - The box is -. = n = ( n ) CHAPTER EQUATIONS AND INEQUALITIES 7

28 6 = n- 6 + = n - + n =. + (x - ) = - + x - 8 = x = x + = - + x = -8 x = 8 x = -. - (x - ) = (x - 6) x + = x x = x x + x = x x = 6x = 6x = 6x 6 = 6 x 6 x = 6. x ( x ) = x 6 x 6 ( x ) = 6 x 6 x x = x - x - 8 = x - x x = x - - x x - 8 = - x = x = x = x = (x - ) = + 0.(x - ) 8-0.x + 0. = + 0.x x = x x +0.x = x+0.x 8. =.8 +0.x = x -.8. =0.x. 0. = 0. x 0. x = (x - ) + x = 7x - (x - ) - - x x = 7x - x x = x + + x - x = x + - x = This is a contradiction. There is no solution.. x + 7 = x - ( - x) x + 7 = x - + x x + 7 = 6x - x x =6x - - x 7 = x = x = x 0 = x x = 0. -x = x -x + x = x + x 0 = x x = y = x 8 + y - 8 = x - 8 y = x - 8 y = x - 8 y = x -. x -( - x)> 8(x - ) + x - + x >8x x - >8x - x - - 8x >8x - - 8x -x - > - -x - + >- + -x >-8 x > 8 x<. ( x ) ( x 6 ) 6 ( x ) ( x 6 ) 6 (x - ) (x - 6) - x - 8 (x - 6) - x - 8 x - - x - 8 x - 6 x x x x -x x -8 8 CHAPTER EQUATIONS AND INEQUALITIES

29 x 8 x 8. V = w ( a c ) V = ( w ( a c )) V = w(a - c) V = wa - wc V +wc = wa - wc + wc V + wc = wa V + wc = wa a = w V + wc w w or a = V w + c. This is solved for w in terms of d and g. 6. w = t + sv w t + s ( w t ) s w t + w - t w w 7. - m > < >m 8. t > -. p is less than or equal to ; p is at most ; the maximum value of p is. CHAPTER EQUATIONS AND INEQUALITIES