Question

1. critical thinking find values for x, y, and z, so that all of the following statements are true: • ( y > x ), ( z < y ), and ( x < 0 ) • ( z + 2 ) and ( z + 3 ) are integers • ( x + z = -2 ) • ( x + y = z ) 38. critical thinking addition and multiplication are said to be closed for whole numbers, but subtraction and division are not. that is, if you add or multiply any two whole numbers, the result is a whole number. which operations are closed for integers? 39. writing in math answer the question that was posed at the of the lesson. how is dividing integers related to multiplying integers? include the following in your answer: • two related multiplication and division sentences, and • an example of each case (same signs, different signs) of dividing integers. 40. on saturday, the temperature fell ( 10^circ ) in 2 hours. which expresses the temperature change per hour? a ( 5^circ ) b ( -2^circ ) c ( -5^circ ) d ( -10^circ ) 41. mark has quiz scores of 8, 7, 8, and 9. what is the lowest score he can get on the remaining quiz to have a final average (mean) score of at least... a 7 b 8 c 9 d 10

Explanation:

Problem 37:

Step 1: Use the equations \(x + z=-2\) and \(x + y = z\)

From \(x + y = z\), we can substitute \(z\) in \(x + z=-2\) to get \(x+(x + y)=-2\), which simplifies to \(2x + y=-2\). Also, we know \(y>x\), \(z < y\), \(x < 0\), and \(z + 2\), \(z + 3\) are integers (so \(z\) is an integer, hence \(x\) and \(y\) are integers too as \(x + y=z\) and \(x+z=-2\)).

Step 2: Let's express \(y\) from \(2x + y=-2\)

We get \(y=-2 - 2x\). Since \(y>x\), substitute \(y\): \(-2-2x>x\), which gives \(-2>3x\), so \(x<-\frac{2}{3}\). And since \(x < 0\) and \(x\) is an integer, possible values for \(x\) start from \(-1,-2,\cdots\)

Step 3: Try \(x=-1\)

If \(x = - 1\), from \(x + z=-2\), we have \(-1+z=-2\), so \(z=-1\). But \(z < y\) and \(x + y=z\) gives \(-1 + y=-1\), so \(y = 0\). Now check \(z < y\): \(-1<0\) (true), \(y>x\): \(0>-1\) (true), \(x < 0\) (true). Also, \(z + 2=-1 + 2 = 1\) (integer), \(z + 3=-1+3 = 2\) (integer). So this works.

Closure property means that when an operation is performed on two elements of a set, the result is also an element of that set. For integers:

• Addition: If we add two integers \(a\) and \(b\) (e.g., \(3+(-2)=1\), \(-5 + 7 = 2\)), the result is an integer. So addition is closed for integers.
• Multiplication: If we multiply two integers \(a\) and \(b\) (e.g., \(4\times(-3)=-12\), \(-2\times(-5)=10\)), the result is an integer. So multiplication is closed for integers.
• Subtraction: If we subtract two integers \(a - b\) (e.g., \(5-7=-2\), \(-3-(-4)=1\)), the result is an integer. So subtraction is closed for integers.
• Division: If we divide two integers, the result may not be an integer (e.g., \(5\div2 = 2.5\), which is not an integer). So division is not closed for integers (except when considering divisibility, but in general, for all integers, division is not closed).

Dividing integers is related to multiplying integers as division is the inverse operation of multiplication.

• Related sentences:
• Multiplication: \(a\times b = c\); Division: \(c\div a=b\) and \(c\div b = a\) (when \(a

eq0\) and \(b
eq0\)).

• Example 1 (same signs, positive): Multiplication: \(3\times4 = 12\); Division: \(12\div3 = 4\) and \(12\div4 = 3\).
• Example 2 (same signs, negative): Multiplication: \((-2)\times(-5)=10\); Division: \(10\div(-2)=-5\) and \(10\div(-5)=-2\).
• Example 3 (different signs): Multiplication: \(3\times(-4)=-12\); Division: \(-12\div3=-4\) and \(-12\div(-4)=3\).

In general, if \(a\) and \(b\) are integers (\(b
eq0\)), then \(a\div b = c\) is equivalent to \(c\times b=a\). The sign rules for division follow from multiplication: same signs give a positive quotient, different signs give a negative quotient, just like in multiplication (same signs give positive product, different signs give negative product).

Answer:

\(x=-1\), \(y = 0\), \(z=-1\) (Note: There might be other solutions, but this is one valid set)

Problem 38: