1) 22 - 2^2 + (-14) / 7 evaluate exponents first 22 - 4 + (-14) / 7 division before add, subtract 22 - 4 + -2 evaluate left-to-right 18 + -2 16 2) b^2 - 4ac = ____ when a=4, b=-4, c=3 (-4)^2 - 4(4)(3) substitute into a, b, c 16 - 48 b^2 implies (-4)^2, not -4^2 -32 3) 7(2r + 6) - 4(2r + 3) switch subtraction to addition 7(2r + 6) + (-4)(2r + 3) distribute 7 and the -4 7(2r) + 7(6) + (-4)(2r) + -4(3) multiply before add 14r + 42 + -8r + -12 combine like terms 14r + -8r + 42 + -12 evaluate left-to-right 6r + 30 ...or... 6(r+5) 4) 8x - (6x - 1) = 2 note: a = b = a + -b 8x + (-1)(6x + -1) = 2 rewrite (switch subtraction to addition) 8x + (-1)(6x) + (-1)(-1)) = 2 distribute the (-1) to (6x + -1) 8x + -6x + 1 = 2 combine like terms 8x and -6x 2x + 1 = 2 8x + -6x = 2x 2x + 1 - 1 = 2 - 1 subtract 1 from both sides 2x = 1 2x/2 = 1/2 divide both sides by 2 x = 1/2 = 0.5 5) (1/4)x - 7 = (1/12)x + 10 (1/4)x - 7 + 7 = (1/12)x + 10 + 7 add 7 to both sides (1/4)x = (1/12)x + 17 (1/4)x - (1/12)x = (1/12)x - (1/12)x + 17 subtract (1/12)x from both sides (3/12)x - (1/12)x = 17 common denominator/equiv. fraction (2/12)x = 17 subtract (12/2)(2/12)x = 17(12/2) multiply both sides by 12/2 x = 17*6 = 102 6) 8% of $267 is _____ 8% = 8/100 = 0.08 0.08 * 267 = $21.36 Jon's commission $218 + $21.36 = $239.36 Jon's salary for the week 7) 8(9x - 4) - 3x > -5(x - 5) + 75x solve for x 8(9x + -4) + -3x > -5(x + -5) + 75x switch subtraction to addition 8(9x) + 8(-4) + -3x > -5(x) + -5(-5) + 75x distribute on both sides 72x + -32 + -3x > -5x + 25 + 75x multiply numbers 69x + -32 > 70x + 25 combine like terms on both sides 69x + -32 + -69x > 70x + 25 + -69x subtract 69x from both sides -32 > x + 25 simplify -32 + -25 > x + 25 + -25 subtract 25 from both sides -57 > x <===|===|===|===)----------//---> -60 -59 -58 -57 -56 -55 8) (4/7)(3a - 7) - (3/4) < 1/4 solve for a (4/7)(3a) - (4/7)(7) - (3/4) < 1/4 distribute the (4/7) on left side (12/7)a - 4 - (3/4) < 1/4 (4/7)*3 = 12/7; (4/7)*7 = 4 (12/7)a - 4 3/4 < 1/4 evaluate -4 - 3/4 (12/7)a - 19/4 < 1/4 convert -4 3/4 to fraction (12/7)a - 19/4 + 19/4 < 1/4 + 19/4 add 19/4 to both sides (12/7)a < 20/4 evaluate right side (12/7)a < 5 (7/12)(12/7)a < 5(7/12) multiply both sides by 7/12 a < 35/12 9) 76% of 1400 points is _____ points 76% = 76/100 = 0.76 0.76 * 1400 = 1064 points needed to pass 1064 - 1006 = 58 points Jon still needs to pass 10) 3x - y + 13 = 0 -3x + 3x - y + 13 = 0 - 3x subtract 3x from both sides -y + 13 = -3x -y + 13 - 13 = -3x - 13 subtract 13 from both sides -y = -3x - 13 -y/-1 = (-3x - 13)/-1 divide both sides by -1 y = -(-3x - 13) y = (-1)(-3x - 13) rewrite; distribute -1 y = (-1)(-3x) + (-1)(-13) y = 3x + 13 11) A = P + Prt solving for t A - P = P + Prt - P subtract P from both sides A - P = Prt (A - P)/Pr = Prt/Pr divide both sides by Pr (A - P)/Pr = t 12) point C is -5 on the x-axis and +1 on the y-axis (-5, 1) point D is +4 on the x-axis and -3 on the y-axis (4, -3) 13) y = 10 - 4x Is (2,2) a solution? x = 2, y = 2 2 = 10 - 4(2) substitute 2 for y and 2 for x 2 = 10 - 8 evaluate the right side 2 = 2 true equation; therefore (2,2) is a solution 14) input -1 repeated (-1,2) (-1,-2) but outputs differ; not a function 15) input are all unique; is a function 16) passes vertical line test; is a function 17) f(x) = x^3 - 2x^2 + 3x + 6 evaluate f(-6) f(x) = (x)^3 - 2(x)^2 + 3(x) + 6 rewrite putting ()s around x f(-6) = (-6)^3 - 2(-6)^2 + 3(-6) + 6 substitute each x with 6 f(-6) = -216 - 2(36) + -18 + 6 expand the exponents f(-6) = -216 - 72 + -18 + 6 evaluate the right side f(-6) = -300 18) f(x) = 21(x - 84) f(x)=42 when x=___ 42 = 21(x - 84) substitute f(x) with 42 42 = 21(x) - (21)(84) distribute the 21 42 = 21x - 1764 evaluate right side 42 + 1764 = 21x - 1764 + 1764 add 1764 to both sides 1806 = 21x 1806/21 = 21x/2 divide both sides by 21 86 = x i.e. f(86) = 42 19) C(d) = 19d + 30 set C(d) = 163 163 = 19d + 30 solve for d 163 - 30 = 19d + 30 - 30 subtract 30 from both sides 133 = 19d 133/19 = 19d/19 divide both sides by d 7 = d i.e. C(7 days) = 163 USD 20) S(x) = 230x + 3800, where x=0 is year 1982 S(1999 - 1982) = 230x + 3800 1999-1982 is x for year 1999 S(17) = 230(17) + 3800 substitute each x with 17 S(17) = 3910 + 3800 230(17)=3910 S(17) = 7710 21) y = -4x + 12 decreasing linear function vi is (0,b) (0,12) hi is (-(b/m),0) (-(12/-4), 0) y-intercept (0,12) x-intercept (3, 0) 22) -x + 2y = 4 1st solve for y +x + -x + 2y = +x + 4 add x to both sids 2y = x + 4 2y/2 = (x + 4)/2 divide both sides by 2 y = x/2 + 2 y = (1/2)x + 2 vi is (0,b) (0,2) hi is (-(b/m),0) (-(2/(1/2), 0) y-intercept (0,2) x-intercept (-4,0) 23) horizontal line implies a constant function; i.e. slope is zero 24) y = (3/4)x - 3 also: y = (3/4)x + -3 vi is (0, b) (0, -3) 25) x = -5 no y; vertical line; not a function; no slope 26) y = x - 4 is also y = 1x - 4 slope = m = 1 27) x = -3 no y; vertical line; not a function; no slope (undefined) 28) 4y - 13x = -3 solve for to find the slope 4y - 13x + 13x = -3 + 13x add 13x to both sides 4y = -3 + 13x 4y/4 = (-3 + 13x)/4 divide both sides by 4 y = (-3/4) + (13/4)x slope = m = 13/4 29) m=3 y-intercept=(0,-2) find y = mx + b y = 3m + -2 (0,-2) is (0,b) 30) y-intercept is when input is 0; therefore, (0, 2.3) 31) find the slope using (0, 2.9) and (1, 3.2) m=(3.2 - 2.9)/(1 - 0) = 0.3/1 = 0.3; y-intercept is (0, 2.3) y = 0.3x + 2.3 32) find y = mx + b y = (-3/8)x + b substitute slope -3/8 into m 2 = (-3/8)4 + b substitute (4, 2) into x and y, respectively 2 = -12/8 + b multiply -3/8 by 4 2 + 12/8 = -12/8 + b + 12/8 add 12/8 to both sides 3 1/2 = b 3 1/2 converted to fraction is 7/2 y = (-3/8)x + 7/2 33) slope of 0 implies constant function; i.e. all inputs produce the same output. The ordered-pair (7, -2) implies an input of 7 produces an output of -2; therefore, y = 0x + -2 or y = -2. 34) (6, 1) (-9, -3) x1 y1 x2 y2 label the points m = (-3 - 1)/(-9 - 6) = -4/-15 = 4/15 find the slope y = (4/15)x + b substitute m=4/15 into y = mx + b 1 = (4/15)6 + b substitute (6, 1) into x and y 1 = 24/15 + b multiply (4/15) by 6 1 - 24/15 = 24/15 + b - 24/15 subtract 24/15 from both sides -9/15 = b -9/15 reduces to -3/5 y = (4/15)x - 3/5 substitute value for b into y = mx + b 35) 2(x + 2) = 2 - 4(x + 2) rewrite switching subtraction into addition 2(x + 2) = 2 + -4(x + 2) distribute both sides 2(x) + 2(2) = 2 + -4(x) + -4(2) 2x + 4 = 2 + -4x - 8 simplify the right side 2x + 4 = -4x - 6 2x + 4 + 4x = -4x - 6 + 4x add 4x to both sides; combine like terms 6x + 4 = -6 6x + 4 - 4 = -6 - 4 subtract 4 from both sides 6x = -10 6x/6 = -10/6 divide both sides by 6 x = -5/3 -10/6 reduces to -5/3 36) y-intercept 10 is (0, 10); x-intercept -4 is (-4, 0) x1 y1 x2 y2 m = (0 - 10)/(-4 - 0) = -10/-4 = 5/2 y = (5/2)x + 10 37) (3, 1158) (9, 1301) x1 y1 x2 y2 m = (1301 - 1158)/(9 - 3) = 143/6 = 23.83 y = 23.83x + b 1158 = 23.83(3) + b 1158 = 71.5 + b 1158 - 71.5 = 71.5 + b - 71.5 1086.5 = b y = 23.83x + 1087 38) (-6, -1) x = -6, y = -1 substitute x and y into 3x + y = -19 3(-6) + -1 = -18 + -1 = -19 true substitute x and y into 4x + 3y = -27 4(-6) + 3(-1) = -24 + -3 = 27 true both are true; therefore, (-6, -1) is a solution 39) start with 5x + y = -11 and solve for y -5x + 5x + y = -5x - 11 subtract 5x from both sides y = -5x - 11 substitute y into the other equation 6x + 5(-5x - 11) = 2 distribute the 5 6x + -25x - 55 = 2 combine like terms -19x - 55 = 2 -19x - 55 + 55 = 2 + 55 add 55 to both -19x = 57 -19x/-19 = 57/-19 divide both sides by -19 x = -3 substitute x into 1st equation 5(-3) + y = -11 solve for y -15 + y = -11 15 + -15 + y = -11 + 15 y = 4 solution: (3, 4) 40) mark: $35/1 hour is a slope; $120 is b sara: $50/1 hour is a slope; $100 is b mark: y = 35x + 120 sara: y = 50x + 100 41) -5x + 2y = -20 multiply both side by 3 3(-5x + 2y) = -20 -15x + 6y = -20 + 7x - 6y = 2 ================= -8x = -18 x = 42) try (3,2): 2(3) + 4(2) = 6 + 8 = 14... 14 ≥ 8 is true <--answer try (0,0): 2(0) + 4(0) = 0 + 0 = 0.... 0 ≥ 8 is false try (1,1): 2(1) + 4(1) = 2 + 4 = 6..... 6 ≥ 8 is false try (1,0): 2(1) + 4(0) = 2 + 0 = 2..... 2 ≥ 8 is false 43) ( 3/4 - 9/16) / (8/5) (12/16 - 9/16) / (8/5) (3/16) / (8/5) (3/16) * (5/8) = 15/128 44) 8 1/5 - 3 1/3 + 1 1/10 41/5 - 10/3 + 11/10 246/30 - 100/30 + 33/30 = (246 - 100 + 33)/30 = 179/30 = 5 29/30 45) (-3x^3)^5 = (-3)^5(x^3)^5 = -243x^(3*5) = -243x^15 46) (-4x^6y^2)^3 = (-4)^3(x^6)^3(y^2)^3 = -64x^(6*3)y^(2*3) = -64x^18y^6 47) (2n/5)^4 = (2n)^4/5^4 = 2^4n^4/5^4 = 16n^4/625 48) 22m^6p^2 22/2 = 11 -------- m^6/m^9 = m^(6-9) = m^-3 = 1/m^3 2m^9p p^2/p^1 = p^(2-1) = p 11 * 1/m^3 * p = 11p/m^3 49) 3x^-2 3y^5z^4 -------- = ------- y^-5z^-4 x^2 50) (x^-2y^6)^-3 = x^(-2*-3)y^(6*-3) = x^6y^-18 = x^6/y^18 51) h(x) = (4x)^-3 h(3) = (4(3))^-3 substitute 3 into x h(3) = 12^-3 = 1/12^3 = 1/1728 exponent law: a^-n = 1/a^n 52) (-5x^4y^4)(-3x^4y^2) -5 * -3 * x^4 * x^4 * y^4 * y^2) 15 * x^(4+4) * y^(4+2) exponent law: a^n * a^m = a^(n+m) 15 * x^8 * y^6 = 15x^8y^6 53) (-2x - 6)(x + 8) ...distribute the -2x, then distribute the -6 (-2x)(x) + (-2x)(8) + (-6)(x) + (-6)(8) -2x^2 - 16x - 6x - 48 combine like terms -2x^2 - 22x - 48 54) (2a - 11)^2 = (2a - 11)(2a - 11) ...distribute the 2a, then distribute the -11 (2a)(2a) + (2a)(-11) + (-11)(2a) + (-11)(-11) 4a^2 - 22a - 22a + 121 combine like terms 4a^2 - 44a + 121 55) 6x^7 - 12x^3 6x^7 12x^3 6 6 ------------ = ----- - ----- = -3 - ------- = -3 + --- -2x^7 -2x^7 -2x^7 (-1)x^4 x^4 6/-2 = -3; x^7/x^7 = 1; 12/-2 = -6; x^3/x^7 = x^(3-7) = x^-4 = 1/x^4