Question

1. critical thinking determine whether the following statement is always true. if it is not, provide a counterexample. if the mathematical operation is defined for all numbers a and b as a b = a + 2b, then the operation * is commutative. 51. writing in math answer the question that was posed at the beginning of the lesson. how is logical reasoning helpful in cooking? include the following in your answer: the hypothesis and conclusion of the statement if you have small, underpopped kernels, then you have not used enough oil in your pan, and examples of conditional statements used in cooking food other than popcorn. 52. grid in what value of n makes the following statement true? if 14n - 12 ≥ 100, then n ≥? 53. if # is defined as #x = \frac{x^{3}}{2}, what is the value of #4? a 8 b 16 c 32 d 64 maintain your skills mixed review simplify each expression. (lesson 1 - 6) 54. 2x + 5y + 9x 55. a + 9b + 6b 56. \frac{3}{4}g+\frac{2}{5}f + \frac{5}{8}g 57. 4(5mn + 6)+3mn 58. 2(3a + b)+3b + 4 59. 6x^{2}+5x + 3(2x^{2})+7x 60. environment according to the u.s. environmental protection agency, a typical family of four uses 100 gallons of water flushing the toilet each day, 80 gallons of water showering and bathing, and 8 gallons of water using the bathroom sink. write two expressions that represent the amount of water a typical family of four uses for these purposes in d days. (lesson 1 - 5) name the property used in each expression. then find the value of n. (lesson 1 - 4) 61. 1(n)=64 62. 12 + 7 = 12 + n 63. (9 - 7)5 = 2n 64. \frac{1}{4}n = 1 65. n + 18 = 18 66. 36n = 0 solve each equation. (lesson 1 - 3) 67. 5(7)+6 = x 68. 7(4^{2})-6^{2}=m 69. p = \frac{22-(13 - 5)}{28\div2^{2}} write an algebraic expression for each verbal expression. (lesson 1 - 1) 70. the product of 8 and a number x raised to the fourth power 71. three times a number n decreased by 10 72. twelve more than the quotient of a number a and 5 getting ready for the next lesson prerequisite skill evaluate each expression. round to the nearest tenth. (to review percents, see pages 802 and 803.) 73. 40% of 90 74. 23% of 2500 75. 18% of 950 76. 38% of 345 77. 42.7% of 528 78. 67.4% of 388

Explanation:

Step1: Recall the definition of commutativity

An operation $*$ is commutative if $a*b = b*a$ for all $a$ and $b$. Given $a*b=a + 2b$, then $b*a=b + 2a$.

Step2: Provide a counter - example

Let $a = 1$ and $b = 2$. Then $a*b=1+2\times2=1 + 4=5$, and $b*a=2+2\times1=2 + 2=4$. Since $a*b
eq ba$ for some values of $a$ and $b$, the operation $$ is not commutative.

Step1: Solve the inequality $14n-12\geq100$

Add 12 to both sides of the inequality: $14n-12 + 12\geq100+12$, which simplifies to $14n\geq112$.

Step2: Isolate $n$

Divide both sides of the inequality $14n\geq112$ by 14: $\frac{14n}{14}\geq\frac{112}{14}$, so $n\geq8$.

Step1: Substitute $x = 4$ into the definition of $\#x$

Given $\#x=\frac{x^{3}}{2}$, when $x = 4$, we have $\#4=\frac{4^{3}}{2}$.

Step2: Calculate the value

First, calculate $4^{3}=4\times4\times4 = 64$. Then $\frac{4^{3}}{2}=\frac{64}{2}=32$.

Step1: Simplify $2x + 5y+9x$

Combine like - terms: $(2x+9x)+5y=11x + 5y$.

Answer:

The statement is not true. A counter - example is when $a = 1$ and $b = 2$, $a*b=5$ and $b*a = 4$.