Questions and Answers #4 Equations and Inequalities

Questions and Answers Sheet 4 Equations and Inequalities Question #1 Help to solve 3−𝑥 𝑥+5 ≥1 Answer: We have: 3−𝑥 𝑥+5 ≥1 We can rewrite it as 3−𝑥 ≥𝑥+5 Subrtract 5 on both sides 3−𝑥−5≥ 𝑥+5−5 −2 − 𝑥 ≥ 𝑥 Add x on both sides −2 − 𝑥 + 𝑥 ≥ 𝑥 + 𝑥 −2 ≥ 2𝑥 −1 ≥ 𝑥 𝑥 ≤ −1 Thus, the answer is 𝑥 ∈ [−1, ∞) Question #2 Help to solve this, please 1 3 2 5 5 6 𝑥+ > 𝑥−1 Answer: 1 3 2 5 5 6 𝑥+ > 𝑥−1 Subtract 1 3 𝑥 3 2 5 from both sides 2 2 5 2 5 5 6 5 𝑥+ − > 𝑥−1− 5 7 6 5 > 𝑥− 5 Subtract 𝑥 3 5 5 7 5 6 6 5 6 6 − 2 𝑥 from both sides − 𝑥> 𝑥− − 𝑥 2𝑥−5𝑥 𝑥 6 3𝑥 6 < 𝑥< 7 ≻ ≻ 5 7 5 7 5 14 5 𝑥 ∈ (−∞, 14 ) 5 Question #3 Help with this, please 2 1 𝑥 3 − 3𝑥 3 − 10 = 0 Answer: We have: 2 1 𝑥 3 − 3𝑥 3 − 10 = 0 Factor it 𝑥 2/3 − 5𝑥 1/3 + 2𝑥 1/3 − 10 = 0 𝑥 1/3 (𝑥 1/3 − 5) + 2(𝑥 1/3 − 10) = 0 (𝑥 1/3 − 5)(𝑥 1/3 + 2) = 0 𝑥 1/3 − 5 = 0 or 𝑥 1/3 + 2 = 0 𝑥 1/3 = 5 or 𝑥 1/3 = −2 𝑥 = 125, 𝑥 = −8 Question #4 Identify: function, equation and inequality 𝑦 = 𝑥2 + 1 2𝑥 − 10 > 𝑥 + 3 𝑥 2 + 𝑦2 = 1 Answer: 1) 𝑦 = 𝑥 2 + 1 There are 2 variables and we will get values of y for each value of x. Thus, it is a function. 2) 2𝑥 − 10 > 𝑥 + 3 Here > sign is used. Thus, it is an inequality. 3) 𝑥 2 + 𝑦 2 = 1 For each values of x or y, we will get 2 values of x and y. Thus, it is an equation. Question #5 𝑛 If 9 = 72 , which of the equations is true? 1) 𝑛 = 72 9 2) 𝑛 = 72 × 9 3) 𝑛 = 9 ÷ 72 4) 𝑛 = 72 + 9 Answer: Given 𝑛 9 = 72 To find which equation is true, consider: 𝑛 = 72 9 𝑛 = 72 × 9 Thus, second option is true. Question #6 Order the expressions by choosing <,>,= 1 −1 ( ) 8 1 −2 ?( ) 8 1 −1 8−2 ? ( ) 8 1 −2 8−2 ? ( ) 8 Answer: 1 −1 1) ( ) 8 1 1 8 ? 1 1 2 6 ( ) 1 −2 ?( ) 8 1 1 8 ? 1 1 64 8?64 8 is less than 64 Thus, 8<64 1 −1 2) 8−2 ? ( ) 8 1 (8)2 1 64 1 64 1 64 ? ? 1 1 (8)1 1 1 8 ?8 is less than 8 1 Thus, 64 <8 1 −2 3) 8−2 ? ( ) 8 1 (8)2 1 64 1 64 1 64 ? ? 1 1 (8)2 1 1 64 ? 64 is less than 64 Thus, 1 64 < 64 Question #7 Provide the solution 1) log 3 (𝑥 2 + 2) = 3 2) log 3 (𝑥 + 4) = log 3 (2𝑥 − 4) 3) log 2 (3𝑥 + 5) ≥ log 2 (𝑥 − 9) 4) log 4 (𝑥 + 1) < log 4 (2𝑥) Answer: Logarithm rules: 𝑏 log𝑏 𝑥 =𝑥 If 𝑏 𝑎 = 𝑥, then log 𝑏 𝑥 = 𝑎 1) log 3 (𝑥 2 + 2) = 3 𝑥 2 + 2 = 33 𝑥 2 + 2 = 27 𝑥 2 = 27 − 2 𝑥 2 = 25 𝑥 = √25 𝑥 = ±5 2) log 3 (𝑥 + 4) = log 3 (2𝑥 − 4) 3log3 (𝑥+4) = 3log3 (2𝑥−4) x+4=2x-4 x-2x=-4-4 -x=-8 x=8 Question #8 9B+C=A, B=? Answer: 9B+C=A 9B=A-C 𝐵= (𝐴−𝐶) is the answer 9 Question #9 Simplify as much as possible (solve for u) 𝑢 6 − 7𝑢 2 = 𝑢 12 Answer: Given: 𝑢 6 − 7𝑢 2 = 𝑢−3(7𝑢) 6 𝑢−21𝑢 6 −20𝑢 6 = = = 𝑢 12 𝑢 12 𝑢 12 𝑢 12 −20𝑢 = 𝑢 2 -40u=u -41u=0 u=0 Question #10 Solve 𝑧 = 3𝑥 − 4𝑦 Answer: z=3x-4y add 4y on both sides 𝑧 + 4𝑦 = 3𝑥 − 4𝑦 + 4𝑦 z+4y=3x 3x=z+4y Divide by 3 both sides 3𝑥 3 = 𝑥= 𝑧+4𝑦 3 𝑧+4𝑦 3 Question #11 Solve 1 ≤ 𝑥 2 + 𝑦2 + 𝑧2 ≤ 4 𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 4, 𝑥 ≥ 0 Answer: (a) Given: 1 ≤ 𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 4 Compare the above inequality with the general equation of a sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑟 2 implies. 𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 4 represents a solid ball with radius 2 centered at origin (0,0,0) . And, 𝑥 2 + 𝑦 2 + 𝑧 2 ≥ 1 represents a solid ball of radius 1 centered at origin. Therefore, the region is, the solid ball of radius 1 removed from a solid ball of radius 2. (b) Given: 𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 4 Compare this inequality with the general equation of a sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑟 2 implies. It represents a solid ball of radius 2 centered at origin. Thus, the required points are the points inside the solid ball. Question #12 Solve 2− 2𝑥−1 𝑥 ≤ 2 2𝑥+1 Answer: 2− 2𝑥−1 𝑥 2𝑥−2𝑥+1 𝑥 1 𝑥 ≤ ≤ ≤ 2 2𝑥+1 2 2𝑥+1 2 2𝑥+1 2𝑥 + 1 ≤ 2𝑥 1≤0 Thus, there is no solution Question #13 Solve equations and inequalities 2𝑥 2 − 11𝑥 − 21 = 0 𝑥2 + 𝑥 ≥ 2 Answer: 1) 2𝑥 2 − 11𝑥 − 21 = 0 2𝑥 2 − 14𝑥 + 3𝑥 − 21 = 0 2𝑥(𝑥 − 7) + 3(𝑥 − 7) = 0 (2𝑥 + 3)(𝑥 − 7) = 0 2x+3=0 or x-7=0 𝑥=− 3 2 or x=7 2) 𝑥 2 + 𝑥 ≥ 2 (𝑥 2 + 𝑥) + (−2) ≥ 2 + (−2) 𝑥2 + 𝑥 − 2 ≥ 0 𝑥 2 − 𝑥 + 2𝑥 − 2 ≥ 0 𝑥( 𝑥2 𝑥 𝑥 2𝑥 𝑥 2 − ) +2( 2 − )≥0 2 𝑥(𝑥 − 1) + 2(𝑥 − 1) ≥ 0 (𝑥 − 1) ( 𝑥(𝑥−1) 𝑥−1 + 2(𝑥−1) 𝑥−1 )≥0 (𝑥 − 1)(𝑥 + 2) ≥ 0 x-1=0 or x+2=0 x-1+1=1 or x+2-2=-2 x=1 or x=-2 𝑥 ≤ −2 −2 ≤ 𝑥 ≤ 1 𝑥≥1 Question #14 Solve equations and inequalities: log(3𝑥 − 1) = log(4 − 𝑥) log 2 𝑥 3 = log 2 (𝑥) ln 8 − 𝑥 2 = ln(2 − 𝑥) log 5 18 − 𝑥 2 = log 5 (6 − 𝑥) Answer: 1)Given log(3𝑥 − 1) = log(4 − 𝑥) 3x-1=4-x 4x=5 𝑥= 5 4 2)Given log 2 𝑥 3 = log 2 (𝑥) 3 log _2^𝑥 = log 2𝑥 (incorrect) Thus, the correct equation is log 2 (𝑥)3 = 3 log 2 𝑥 3) Given ln 8 − 𝑥 2 = ln(2 − 𝑥) Remove ln on both sides 8 − 𝑥2 = 2 − 𝑥 𝑥2 − 𝑥 − 6 = 0 (x-3)(x+2)=0 x=3 or x=-2 Question #15 Solve equations and inequalities: 3(2x-4)=-2(3x-1) 1 3 1 (9𝑥 − 6) = (6𝑥 + 2) 2 4+5(2-x)=1-2(2x+1) Answer: 1) 3(2x-4)=-2(3x-1) 6x-=-6x+2 12x=14 𝑥= 2) 1 3 14 12 = 7 6 1 (9𝑥 − 6) = (6𝑥 + 2) 2 18x-12=18x+6 Hence, there is no solution Question #16 Solve for 𝑑2 , please: 5(𝑑1 −𝑑2 ) 𝐿 =𝑊 Answer: We have: 5(𝑑1 −𝑑2 } 𝐿 =𝑊 Simplify and cross multiply 5𝑑1 − 5𝑑2 = 𝑊𝐿 ⇒ 5𝑑1 − 𝑊𝐿 = 5𝑑2 ( 5𝑑1 −𝑊𝐿 5 𝑑2 = ) = 𝑑2 5𝑑1 −𝑊𝐿 5 Question #17 Solve for s −6𝑟 + 16𝑥 = −4 Answer: Given: -6r+16x=-4 Put all the variables, constant except s on the right hand side 16s=-4+6r Divide by 16 𝑠= 𝑠= 𝑠= −4 + 6𝑟 16 2(−2+3𝑟) 16 3𝑟−2 8 (answer) Question #18 Solve for x (a) 82−𝑥 = 2 1 𝑥 1 2 8 (b) ( ) < (c) 5𝑥 = 25𝑥−2 (d) 3𝑥+2 ≥ 27 (e) 43𝑥 = 8𝑥−1 Answer: a) 82−𝑥 = 2 (23 )2−𝑥 = 2 26−3𝑥 = 21 6-3x=1 6-1=3x 5 3 =𝑥 1 𝑥 1 2 8 b) ( ) < 1 2𝑥 1 < 23 23 < 2 𝑥 33 c) 5𝑥 = 25𝑥−2 5𝑥 = (52 ) 𝑥−2 5𝑥 = 52𝑥−4 x=2x-4 4=2x-x x=4 Question #19 Solve for x using quadratic formula 3𝑥 2 + 6𝑥 − 2 = 0 Answer: Given: 3𝑥 2 + 6𝑥 − 2 = 0 Using quadratic formula 𝑥= −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎 Thus, 𝑥= = = −6±√62 −4(3)(−2) 2(3) −6±√36+24 6 −6±√60 𝑥1 = 6 −6+√60 6 or 𝑥2 = −6−√60 6 𝑥1 ≈ 0.30 or 𝑥2 ≈ −2.30 Question #20 Solve for x using quadratic formula 6𝑥 2 + 2𝑥 = 3 Answer: 6𝑥 2 + 2𝑥 = 3 6𝑥 2 + 2𝑥 − 3 = 0 𝑥= = = = = = −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎 −2±√22 −4(6)(−3) 2(6) −2±√4+72 12 −2±√76 12 −2±2√19 12 −2±√19 𝑥= 6 −2+√19 6 ,𝑥 = −2−√19 6