Questions and Answers #1 Equations and Inequalities

Questions and Answers Sheet 1 Equations and Inequalities Question #1 Prove that ||𝑥| − |𝑦|| ≤ |𝑥 − 𝑦| I've seen the full proof of the Triangle Inequality |𝑥 + 𝑦| ≤ |𝑥| + |𝑦|. However, I haven't seen the proof of the reverse triangle inequality: ||𝑥| − |𝑦|| ≤ |𝑥 − 𝑦|. Would you please prove this using only the Triangle Inequality above? Thank you very much. Answer: |𝑥| + |𝑦 − 𝑥| ≥ |𝑥 + 𝑦 − 𝑥| = |𝑦| |𝑦| + |𝑥 − 𝑦| ≥ |𝑦 + 𝑥 − 𝑦| = |𝑥| Move |x| to the right hand side in the first inequality and |y| to the right hand side in the second inequality. We get |𝑦 − 𝑥| ≥ |𝑦| − |𝑥| |𝑥 − 𝑦| ≥ |𝑥| − |𝑦|. From absolute value properties, we know that |𝑦 − 𝑥| = |𝑥 − 𝑦|, and if 𝑡 ≥ 𝑎 and 𝑡 ≥ −𝑎 then 𝑡 ≥ |𝑎|. Combining these two facts together, we get the reverse triangle inequality: |𝑥 − 𝑦| ≥∣∣ |𝑥| − |𝑦| ∣∣. Question #2 Solve the equation 1/(3𝑥 − 5)2 − 25 = 0 Answer: 1/(3𝑥 − 5)2 − 25 = 0 1/(3𝑥 − 5)2 = 25 1/25 = (3𝑥 − 5)2 √(1/25) = 3𝑥 − 5 ±1/5 = 3𝑥 − 5 5 ± 1/5 = 3𝑥 (25 ± 1)/5 = 3𝑥 (25 ± 1)/15 = 𝑥 𝑥 = 8/5; 𝑥 = 26/15 Question #3 Find x and y, where 7𝑥 + 4𝑦 = 5; and −𝑥 − (4/7)𝑦 = −5/7; Answer: Given: (a) 7𝑥 + 4𝑦 = 5 (b) −𝑥 − (4/7)𝑦 = −5/7 Multiply (b) by 7. −7𝑥 − 4𝑦 = −5 Add (a) and 7(b) 7𝑥 + 4𝑦 − 5 − 7𝑥 − 4𝑦 + 5 = 0 Since the equation is true, there's an infinite number of solutions Question #4 Given: A parallelogram XYZW, (𝑌𝑍) = 2𝑏, (𝑍𝑊) = 𝑎 + 2, (𝑊𝑋) = 𝑏 + 1, (𝑋𝑌) = 3𝑎 − 4 Answer: In a parallelogram the opposite sides are congruent. Thus, 2b=b+1 and a+2=3a-4 Solve each equation individually 2b=b+1 b=1 a+2=3a-4 -2a=-6 a=3 Question #5 (12𝑥 − 10𝑦)/(3𝑥 + 2𝑦) × (6𝑥 + 4𝑦)/(10𝑦 − 12𝑥) =? Answer: (12𝑥 − 10𝑦)/(3𝑥 + 2𝑦) × (6𝑥 + 4𝑦)/(10𝑦 − 12𝑥) = (2(6𝑥 + 5𝑦))/(3𝑥 + 2𝑦) × (2(3𝑥 + 2𝑦))/(−2(6𝑥 + 5𝑦)) = (6𝑥 + 5𝑦)/(6𝑥 + 5𝑦) × (3𝑥 + 2𝑦)/(3𝑥 + 2𝑦) × (2 × 2)/−2 = 1 × 1(−2) = −2 Question #6 A wooden model of a building has a scale of 1:360. The wooden model is 35(2/5) cm high. How high is the building in m? Answer: Let the building's height be h. (35(2/5))/ℎ = 1/360 35(2/5)(360) = ℎ ℎ = 177/5(360) ℎ = 12744𝑐𝑚 12744𝑐𝑚 = 12744/100 𝑚 = 127,44𝑚 The building is 127,44m high. Question #7 √(3𝑦 + 2) = 0 Find y Answer: √(3𝑦 + 2) = 0 Only 02 = 0 , so 3𝑦 + 2 = 0 3𝑦 = −2 𝑦 = −2/3 Question #8 Solve the following equation: 24/15𝑥 + 2 = 34/15 Answer: 24/15𝑥 + 2 = 34/15 15(24/15𝑥 + 2) = 15(34/15) 24𝑥 + 30 = 34 24𝑥 = 4 𝑥 = 4/24 𝑥 = 1/6 Question #9 Solve the inequality: 3𝑥 3 − 3𝑥 < 4𝑥 2 − 4 Answer: 3𝑥 3 − 3𝑥 < 4𝑥 2 − 4 3𝑥 3 − 3𝑥 + 4 < 4𝑥 2 3𝑥 3 − 3𝑥 + 4 − 4𝑥 2 < 0 Identify the intervals that satisfy the condition: <0 𝑥 < −1; 1 < 𝑥 < 4/3 Question #10 Solve the equation: 4 √4 + 8𝑥 = 2𝑥 1/4 Answer: 4 √4 + 8𝑥 = 2𝑥 1/4 4 4 ( √4 + 8𝑥 ) = (2𝑥 (1/4) ) 4+8x=16x 8x=4 x=0.5 4 Question #11 √(3𝑥) + √12 = (𝑥 + 5)/√3 Find x Answer: √(3𝑥) + √12 = (𝑥 + 5)/√3 (√(3𝑥) + √12) (√3) = (𝑥 + 5)/√3(√3) √(9𝑥) + √36 = 𝑥 + 5 3𝑥 + 6 = 𝑥 + 5 2𝑥 = −1 𝑥 = −0.5 Question #12 Simplify the following expression: 7 × (10 + 𝑥) Answer: Given: 7 × (10 + 𝑥) Multiply: 7 × (10 + 𝑥) = (7 × 10) + (7𝑥) = 70 + 7𝑥 Question #13 Find x and y, if (6 + 𝑖)(𝑥 + 𝑦𝑖) = 2 and, 𝑖 2 = −1 Answer: (6 + 𝑖)(𝑥 + 𝑦𝑖) = 2 6𝑥 + 𝑥𝑖 + 6𝑦𝑖 + 𝑦𝑖 2 = 2 Substitute 𝑖 2 = −1: 6𝑥 + 𝑥𝑖 + 6𝑦𝑖 − 𝑦 = 2 (6𝑥 − 𝑦) + (𝑥 + 6𝑦)𝑖 = 2 Since 2=2+0i, it's appropriate to equate the real parts and the imaginary parts of each side to obtain 2 equations: 6x-y=2 (a) x+6y=0 (b) Multiply (a) by 6: 36x-6y=12 (c) Add (b) and (c) to solve for x: 37x=12 x=12/37 Use (a) to solve for y: 6 × 12/37 − 𝑦 = 2 −𝑦 = 2 − 72/37 𝑦 = −(2/37) Question #14 Find x, if −5 + (𝑥/3) = 0 Answer: −5 + (𝑥/3) = 0 𝑥/3 = 5 𝑥 =5 ×3 x=15 Substitute to verify −5 + 15/3 = 0 −5 + 5 = 0 0=0 Question #15 12 = 12/6 + 12/𝑦 Solve the equation to find y. Answer: 12 = 12/6 + 12/𝑦 12/6 = 2, so 12 = 2 + 12/𝑦 12/𝑦 = 12 − 2 𝑦 = 12/10 𝑦 = 6/5 Substitute y=6/5 to the given equation and verify: 12 = 12/6 + 12/(6/5) 12 = 12/6 + (12 × 5)/6 12 = 12/6 + 60/6 12 = 2 + 10 12 = 12 Question #16 Find all solutions of the given system of equations (6𝑥, −,5𝑦, = ,7), (5𝑥, −,6𝑦, =, −7): Answer: (6𝑥, −,5𝑦, = ,7), (5𝑥, −,6𝑦, =, −7): I keep the 0x and 0y for you to better see what's going on. (6𝑥, −,5𝑦, = ,7), (0𝑥, −,11𝑦/6, =, −77/6): (6𝑥, −,5𝑦, = ,7), (0𝑥, +, 𝑦, = ,7): (6𝑥, −,5 ∗ 7, = ,7), (0𝑥, +, 𝑦, = ,7): (6𝑥, −,35, = ,7), (0𝑥, +, 𝑦, = ,7): (6𝑥, +,0𝑦, = ,42), (0𝑥, +, 𝑦, = ,7): (𝑥, +,0𝑦, = ,7), (0𝑥, +, 𝑦, = ,7): 𝑦 = 7, 𝑥 = 7 Question #17 How do you graph the inequality 𝑦 < 2𝑥 − 1 ? Answer: Find the x- and y-intercepts. X-intercept: value of x when y=0 Substitute an equal sign for the less than symbol. y=2x-1 Substitute 0 for y 0=2x-1 Add 1 to both sides. 1=2x Divide both sides by 2. 1 =𝑥 2 1 The x-intercept is ( , 0 ). 2 Y-intercept: value of y when x=0 Substitute 0 for x . 𝑦 = 2(0) − 1 𝑦 = −1 The y-intercept is ( 0, -1 ) Question #18 Describe the sets of points in space whose coordinates satisfy the inequality or combination of equations and inequalities. 1 ≤ 𝑥 2 + 𝑦2 + 𝑧2 ≤ 4 Answer: The given combinations of equations and inequalities is 1 ≤ 𝑥 2 + 𝑦2 + 𝑧2 ≤ 4 Write the standard equation for the sphere of radius a and center (𝑥0 , 𝑦0 , 𝑧0 ) is (𝑥 − 𝑥0 )2 + (𝑦 − 𝑦0 )2 + (𝑧 − 𝑧0 )2 = 𝑎2 The equation (𝑥 − 0)2 + (𝑦 − 0)2 + (𝑧 − 0)2 = 1 Compare the equations, the center C is (0,0,0) and radius a is 1. The equation (𝑥 − 0)2 + (𝑦 − 0)2 + (𝑧 − 0)2 = 22 Compare the equations, the center C is (0,0,0) and radius a is 2. For 1 ≤ 𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 4 in the Cartesian coordinate system represents that the solid sphere enclosed between the radius 1 and radius 2 which centered at the origin (0,0,0) . Hence, the set of points in space whose coordinates satisfy the given combinations 1 ≤ 𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 4 is the solid enclosed between the sphere of radius 1 unit and 2 unit centered at the origin. Question #19 Please, solve for the values of x for each of the following exponential equations and inequalities. (a) 82−𝑥 = 2 (b) (1/2)𝑥 < (1/8) (c) 5𝑥 = 25𝑥−2 (d) 3𝑥+2 ≥ 27 (e) 43𝑥 = 8𝑥−1 Answer: I did only three first a) 82−𝑥 = 2 8 = 23 23(2−𝑥) = 2 So 3(2 − 𝑥) = 1 6 − 3𝑥 = 1 3𝑥 = 6 − 1 𝑥= 5 3 1 𝑥 1 2 8 b) ( ) < ( ) 1 2𝑥 < 1 8 8 < 2𝑥 2𝑥 > 8 8 = 23 2 𝑥 > 23 Thus, x>3 c) 5𝑥 = 25𝑥−2 25 = 52 5𝑥 = (5)2(𝑥−2) So, 𝑥 = 2(𝑥 − 2) 𝑥 = 2𝑥 − 4 2𝑥 − 𝑥 = 4 𝑥=4 Question #20 Please, describe the sets of points in space whose coordinates satisfy the given inequality. 𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 1 Answer: Consider the information we have 𝑥 2 + 𝑦 2 + 𝑧 2 = 1 represent a sphere with radius 1 and center at the point (0,0,0) . So that the inequality 𝑥 2 + 𝑦 2 + 𝑧 2 ≤ 1 represents all the points inside the sphere.