Questions and Answers #15 Equations, Expressions and Inequalities

Questions and Answers Sheet 15 Equations, Expressions and Inequalities Question #1 How do you simplify 6-4c+c ? Answer: Step 1 Given: 6-4c+c Gather like terms. 6+(-4c+c) Step 2 Combine like terms. 6+(-3c) Simplify. 6-3c. Question #2 How do you simplify −𝑦 + 9𝑧 − 16𝑦 − 25𝑧 ? Answer: To simplify this, we have to combine like terms. First, let's color code the like terms: -y+9z-16y-25z Now combine like terms: Hence, -17y-16z Question #3 How do you simplify -7s-5-7s+9 ? Answer: Combine like terms. -7s-7s-5+9 Simplify: -14s+4 Question #4 How do you simplify 5c-3d-12c+d ? Answer: Solution: 5c-3d-12c+d We group the like terms together: (5c-12c)+(-3d+d) =-7c-2d Question #5 How do you simplify (𝑥 2 + 7𝑥 − 4) + (𝑥 2 + 9) ? Answer: Step 1 There is no multiplication or division of the brackets so you simply remove the brackets, collect like terms and do the arithmetic (math) Removing the brackets 𝑥 2 + 7𝑥 − 4 + 𝑥 2 + 9 Step 2 An extra thought: If we had been given (𝑥 2 + 7𝑥 − 4) − (𝑥 2 + 9) Then the minus sign in front of (𝑥 2 + 9) would mean that we have to change the sign of everything inside that right hand bracket. Suppose that had been the case, then we would have had: 𝑥 2 + 7𝑥 − 4 − 𝑥 2 − 9 instead. Step 3 Collecting terms: 𝑥 2 + 𝑥 2 + 7𝑥 + 9 − 4 Answer: 2𝑥 2 + 7𝑥 + 5 Question #6 3 1 7 1 4 3 8 2 How do you simplify − − + 𝑥 − ? Answer: Explanation: This is no different from any other fractions being added. Find the LCD and make equivalent fractions. 3 1 7 1 − − + 𝑥− = = 4 3 8 2 −18−8+21𝑥−12 24 21𝑥−38 24 Question #7 What is x if 2 𝑥 − 3𝑥 = −4 ? Answer: Step 1 2 𝑥 − 3𝑥 = −4 2 − 3𝑥 2 = −4𝑥 3𝑥 2 − 4𝑥 − 2 = 0 which is in the standard quadratic form: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 where b=-4, a=3, c=-2 Step 2 Hence, the roots of the equation are given by: 𝑥= −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎 𝑥=( 𝑥= 𝑥= 4±√16+24 6 4±√40 ) 6 2±√10 3 Question #8 How do you simplify 2x-5-4x+8 ? Answer: Given: 2x-5-4x+8 Re-arrange to group terms with common variables 2x-4x-5+8 =-2x+3 Question #9 How do you solve for x? 1 𝑒 −2𝑥 + 𝑒 𝑥 = ? 3 Answer: Collect 𝑒 𝑥 on the left: 1 𝑒 𝑥 (𝑒 −2 + 1) = rearrange: 𝑒𝑥 = 3 1 3(𝑒 −2 +1) apply the ln on both sides: ln[𝑒 𝑥 ] = ln [ so: 𝑥 = ln [ 1 ] 3(𝑒 −2 +1) 1 ] 3(𝑒 −2 +1) use the properties of log relating sums and subtractions to multiplications and divisions to get: Therefore, 𝑥 = ln(1) − ln[3(𝑒 −2 + 1)] = 0 − ln(3) − ln(𝑒 −2 + 1) Question #10 How do you simplify 6a+2a ? Answer: A multiplication is just a short way to write a sum, in the sense that 6𝑎 = ⏟ 𝑎 + 𝑎 + 𝑎 + 𝑎 + 𝑎 + 𝑎 , and 6 2𝑎 = ⏟ 𝑎 + 𝑎. 2 Now, let's add the two terms: 6𝑎 + 2𝑎 = ⏟ 𝑎+𝑎+𝑎+𝑎+𝑎+𝑎+⏟ 𝑎+𝑎 ⏟ 6 2 8 which means that the total is eight times a. Question #11 What is x if 5x-14=21 ? Answer: Solution: 5x-14=21 Add 14 to both sides. 5x-14+14=21+14 5x=35 Divide both sides by 5. 5𝑥 35 = 5 5 Answer: x=7 Question #12 How do you simplify (4𝑎 − 𝑏) + (3𝑎 − 5𝑏) ? Answer: Explanation: (4a-b)+(3a-5b) 4a-b+3a-5b Grouping like terms together: 3a+3a-b-5b =7a-6b Question #13 What is x if 𝑥 − 3 𝑥2 + 14 = 8 ? Answer: 𝑥− 3 𝑥2 + 14 = 8 Subtract 8 from each side: 3 𝑥− 2+6=0 𝑥 Multiply each side by 𝑥 2 : (𝑥 2 ) (𝑥 − 3 𝑥2 + 6) = 0 Distribute and simplify: 𝑥 3 − 3 + 6𝑥 2 = 0 0 = 𝑥 3 + 6𝑥 2 − 3 From here, I think the best option to solve this would be to use a graphing calculator. On the TI-84 plus, I used the numeric solver. 𝑥 = −0.671, 𝑥 = 0.756, 𝑥 = 5.914 Question #14 2 What is x if 10 = − (4𝑥 + 5) ? 3 Answer: Step 1 2 10 = − (4𝑥 + 5) 3 30=-2(4x+5) 30=-8x-10 30-10=-8x 20=-8x Step 2 20 −8 5 −2 =𝑥 =𝑥 Question #15 3 What is x if 4𝑥 + − 9 = 5? 𝑥 Answer: 4x+3/x-9=5 multiplying both the sides with x x(4x+3/x-9)=5.x 4x.x+3/x.x-9.x=5.x 4𝑥 2 + 3 − 9𝑥 = 5𝑥 Subtracting 5x from both sides 4𝑥 2 + 3 − 9𝑥 − 5𝑥 = 5𝑥 − 5𝑥 4𝑥 2 + 3 − 14𝑥 = 0 4𝑥 2 − 14𝑥 + 3 = 0 Apply the quadratic formula. 4𝑥 2 − 14𝑥 + 3 is in the form of a quadratic equation 𝑎2 𝑥 + 𝑏𝑥 + 𝑐 , where a=4, b=-14, and c=3 . Quadratic Formula 𝑥= 𝑥= 𝑥= 𝑥= −𝑏±√𝑏 2 −4𝑎𝑐 = 2𝑎 −(−14)±√(−14 2 )−(4⋅4⋅3) 2⋅4 14±√196−48 = 8 14±√148 8 Solve for x. 𝑥= 14+√148 14−√145 8 , 8 Question #16 3 What is x if 2𝑥 − = 11 ? 𝑥 Answer: Step 1 3 2𝑥 − = 11 𝑥 Multiply both sides by the LCD x. 2 x( x) − 3 ( x ) = 11( x) = x 2𝑥 2 − 3 = 11𝑥 Step 2 Move all terms to the left side. 2𝑥 2 − 3 − 11𝑥 = 0 Rearrange the terms. 2𝑥 2 − 11𝑥 − 3 Step 3 Apply the quadratic formula. 2𝑥 [2} − 11𝑥 − 3 is in the form of a quadratic equation ax+bx+c , where a=2, b=-11, and c=-3 . Quadratic Formula −𝑏 ± √𝑏 2 − 4𝑎𝑐 𝑥= = 2𝑎 −(−11) ± √(−112 ) − (4 ⋅ 2 ⋅ −3) 𝑥= = 2⋅2 11±√121+24 𝑥= = 𝑥= 4 11±√145 4 Solve for x. 11 + √145 11 − √145 𝑥= , 4 4 Question #17 What is x if 5x-2=2x+7 ? Answer: 5x-2=2x+7 5x-2x=7+2 3x=9 9 𝑥= 3 x=3 Question #18 2 1 3 3 What is x if 4 = 𝑥 + 9 − 𝑥 ? Answer: Given: 2 1 4= 𝑥+9− 𝑥 3 3 Multiply everything by 3 to remove the denominator 2 1 (4 ⋅ 3) = ( ⋅ 3 ⋅ 𝑥) + (9 ⋅ 3) − ( ⋅ 3 ⋅ 𝑥) 3 3 Rewrite the equation 12=2x+27-1x Collect like terms 12=1x+27 Isolate x -15=1x Answer: x=-15 Question #19 How do you evaluate 2𝑦 + 9𝑥 − 7 when 𝑦 = 3 and 𝑥 = 5 ? Answer: Answer: 44 Explanation: 2𝑦 + 9𝑥 − 7 is the expression provided to us. The values for the variables are 𝑦 = 3 and 𝑥 = 5 We substitute the values of the variables provided into the expression: 2𝑦 + 9𝑥 − 7 = 2 ⋅ 3 + 9 ⋅ 5 − 7 = 6 + 45 − 7 Hence, 44. Question #20 What is x if 5x+4-8x=13 ? Answer: Step 1 Given: 5x+4-8x=13 . Add like terms. -3x+4=13 Step 2 Subtract 4 from each side. -3x=13-4 =9 Step 3 Divide by -3. 9 𝑥= −3 =-3 Therefore, x=-3