Questions and Answers #12 Equations, Expressions and Inequalities

Questions and Answers Sheet 12 Equations, Expressions and Inequalities Question #1 5 How do you translate " 𝑘 6 " into a verbal expression? Answer: We'll want to get this in a form involving a root first. 𝑎 𝑏 Recall that 𝑥 𝑏 = √𝑥 𝑎 5 6 𝑘 6 = √𝑘 5 Now, we can translate this as "The sixth root of a number raised to the fifth power" Question #2 How do you simplify 3(𝑥 + 4) − 6 ? Answer: First target the expression inside the parentheses, 𝑥 + 4 . This cannot be simplified inside the parentheses so move out the parentheses to complete the distribution of the factor 3, which is also implied multiplication. 3(x+4)-6 3x+12-6 Combine Like Terms. (CLT) Answer: 3x+6 Question #3 1 How do you use the binomial theorem to approximate 1.082 and hence find √3 to 4 significant figures? Answer: Cutting the binomial series short, 1 𝑥 𝑥2 2 8 (1 + 𝑥)2 ≅ 1 + − 5 Then √3 = √1.08 ≅ 1 + 3 Explanation: 1 0.08 2 − 0.082 8 = 1.732 1 1 1 1 1 1 1 1 2 2! 2 2 3! 2 2 2 (1 + 𝑥)2 = 1 + ( ) 𝑥 + ( ) ( ) ( − 1) 𝑥 2 + ( ) ( ) ( − 1) ( − 2) 𝑥 3 + 𝑥 𝑥2 2 8 =1+ − + 𝑥3 16 − If x is small this will converge rapidly So with x=0.08 we get: 1 √1.08 = (1 + 0.08)2 ≅ 1 + 0.08 2 − 0.082 8 = 1 + 0.04 − 0.0008 = 1.0392 Now 1.08 = 33 52 33 3 2 3 So √1.08 = √ 2 = √3 ⋅ ( ) = √3 5 5 5 5 5 √3 = 3 √1.08 ≅ 3 ⋅ 1.0392 = 1.732 Question #4 3 1 3 1 3 If √3 ( √𝑥 − 3 ) = 2, then √𝑥 + 3 = what? √𝑥 √𝑥 Answer: Step 1 We start with the original function: 3 1 3 √3 ( √𝑥 − 3 ) = 2 √𝑥 and we want to solve for: 1 3 √𝑥 + 3 (note the + instead of -) √𝑥 which means we need to do a bit more work and solve for x and then use our x value to solve for: 1 3 ( √𝑥 + 3 ) √𝑥 We then take each side to the exponent 3 (which gets rid of the cube root on the left hand side): 3 3 1 3 ( √3 ( √𝑥 − 3 )) = 23 √𝑥 Step 2 Taking the cube root of a number is equivalent to taking a fractional exponent of 1/3 and so this simplifies to: 3 3 1 3 (3 ( √𝑥 − 3 )) = 8 √𝑥 1 3 3 ( √𝑥 − 3 ) = 8 √𝑥 We then divide both sides by 3 (to get ride of the multiplication by 3 on the left hand side) 1 3 3 √𝑥− 3 √𝑥 3 = 8 1 3 8 √𝑥 3 3 √𝑥 − 3 = 3 Next we multiply both sides by √𝑥 to remove it from the denominator of the second term 3 2 3 2 3 √𝑥 − 1 = √𝑥 − 8⋅ 3√𝑥 3 8⋅ √𝑥 3 −1 =0 You might not immediately recognize it, but this is a quadratic equation. To make that more obvious, 3 3 we're going to temporarily substitute √𝑥 . Let's choose a new variable y to represent √𝑥 : 3 𝑦 = √𝑥 And now our equation becomes: 𝑦 𝑦2 − 8 − 1 = 0 3 which more closely resembled what we expect from a quadratic equation. Using the quadratic formula and the letter y now instead of the traditional x: 𝑦= −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎 we find that: 𝑦 = 3 and 𝑦 = − 1 3 Step 3 3 Now we'll substitute back our function of x for y: 𝑦 = √𝑥 3 √𝑥 = 3 𝑥 = 33 x=27 1 3 and √𝑥 = − 3 1 3 𝑥 = (− ) 𝑥=− 3 1 27 So we have two solutions for x: 1 𝑥 = 27 and 𝑥 = − 27 1 Using either of these values (27 or − ) , we should be able to find the answer to the original problem: 27 1 3 √𝑥 + 3 √𝑥 3 = √27 + 3 1 √27 3 We know that √27 = 3 which means: 1 =3+ = = 9+1 3 3 10 3 Question #5 How do you simplify 15𝑑 − 9 + 2𝑑 ? Answer: 15d-9+2d =15d+2d-9 =(15+2)d-9 =17d-9 Answer: Regroup to get 17d-9 . Question #6 How do you solve 15a+6d-12a-8b ? Answer: The expression can be simplified by grouping like terms: 15a+6b-12a-8b =(15a-12a)+6b-8b Answer: =3a-2b Question #7 How do you simplify 24b-b ? Answer: Factor out the GCF b. Explanation: 24b-b= b(24-1)= b(23)= 23b Question #8 How do you simplify 4𝑥 2 − 20𝑥 + 32𝑥 + 1 + 2𝑥 2 − 25𝑥 − 274𝑥 + 2 ? Answer: Answer: 6𝑥 2 − 287𝑥 + 3 Explanation: 4𝑥 2 − 20𝑥 + 32𝑥 + 1 + 2𝑥 2 − 25𝑥 − 274𝑥 + 2 Grouping all the coefficients of 𝑥 2 , x and the constant terms together. = 4𝑥 2 + 2𝑥 2 − 20𝑥 + 32𝑥 − 25𝑥 − 274𝑥 + 2 + 1 = 6𝑥 2 − 287𝑥 + 3 Question #9 How do you simplify 12s-4t+7t-3s ? Answer: 12s-4t+7t-3s=9s+3t or =3(3s+t) Explanation: 12s-4t+7t-3s can be regrouped as (12s-3s)+(7t-4t) =(12-3)s+(7-4)t =9s+3t or, extracting a common factor of 3 =3(3s+t) Question #10 How do you simplify the expression 15r-16+3r ? Answer: 15r-16+3r=18r-16 Explanation: Simplify: 15r-16+3r Collect like terms. (15r+3r)+(-16) Combine like terms. 18r-16 . Question #11 How do you simplify 2x+5(6x-4)+1 ? Answer: 2𝑥 + 5 ⋅ 6𝑥 + 5 ⋅ (−4) + 1 2𝑥 + 30𝑥 − 20 + 1 32𝑥 − 19 Question #12 How do you simplify 2x-8+6x-8 ? Answer: Step 1 8x-16=0 8x=16 Step 2 𝑥= 16 8 Answer: x=2 Question #13 How do you simplify (5m+5n)+(6m-4n) ? Answer: First, remove all of the terms from parenthesis. Be careful to handle the signs of each individual term correctly: 5m+5n+6m-4n Next, group like terms: 5m+6m+5n-4n Now, combine like terms: (5+6)m+(5-4)n 11m+1n 11m+n Question #14 How do you simplify 8c+11-6c ? Answer: First, group the like terms: 8c-6c+11 Combine the like terms: (8 − 6)𝑐 + 11 2c+11 Question #15 How do you simplify 2-3[x-2-5(x-1)]? Answer: First multiply -5: 2-3[x-2-5x+5]= add similar terms inside the square brackets: =2-3[3-4x]= multiply -3: =2-9+12x= Hence, =12x-7 Question #16 4 𝑥−4 7 6 4 𝑥−4 7 6 How do you translate "𝑥 + = 2 " into a verbal expression? Answer: This equation which is 𝑥 + = 2 Can be verbally expressed as: X plus 4 by 7 equals to 2 times x minus 4 upon 6 OR X plus 4 by 7 equals to 2 into x minus 4 upon 6 Upon and by can also be replaced by themselves if you like to. Question #17 1 How to solve 2 𝑥 𝑥2 3 4 + + 1 where 𝑥 = ? 2 Answer: Given: 1 2 1 2 𝑥 𝑥2 3 4 1 2 ( ) + + + 3 + with 𝑥 = 2 1 2 2 4 1 1 2 2 1 3 4 1 1 2 1 2 1 2 6 = + 1 + 1 4 1 1 4 4 = + ⋅ +( )⋅ 3 = + + 1 16 least common multiple is 3 × 16 = 48 1 1 1 + + 2 = = 6 24 48 35 + 64 8 48 + 35 48 48 Question #18 How do you simplify 3𝑥 + 1𝑥 + 2 − (−5𝑥) + 7 ? Answer: We can add everything together. The variables for themselves and the integers for themselves. 3x+1x+2-(-5x)+7 4x+2-(-5x)+7 For -(-5x) you know that (-) ⋅ (-)=+ . So, since it's written as -(-5x) it will be equal to 5x. 4x+2+7-(-5x) 4x+9+5x We will then land on the answer, Therefore, 9x+9 Question #19 How do you simplify the expression -2b+b ? Answer: Answer: -b Explanation: One could take the common factor out and factorise to get: b(-2+1) Add the terms in brackets. = 𝑏 ⋅ −1 =-b Question #20 How do you simplify 3a+5a+5p? Answer: To simplify this, we have to combine like terms. First, let's color code the like terms: 3a+5a+5p As you can see, 3a and 5a are like terms, so we can add them together: 8a Now, we still have 5p left, because that is not a like term. So we just leave that alone and our final answer is: 8a+5p