Question

1. use inductive reasoning to make a conjecture about the sum of two negative integers. then use deductive reasoning to show that the conjecture is true.
2. solve -3(6x - 1)=6x - 9. justify each step.

in exercises 27 and 28, name the property that the statement illustrates.

1. (overline{jk}=overline{jk})
2. (angle c=angle c)
3. modeling real life a fitness center charges members an initial fee of $10 and a monthly fee of $21.99. find the total cost of 1 year of membership.
4. sketch a diagram showing (overline{ab}) intersecting (overline{cd}) at point (k), so that (ak = kb) and (ck = kd).

Explanation:

25.

Step1: Inductive reasoning for conjecture

Let's take some examples. If we have - 2 and - 3, -2+( - 3)=-5; if we have -5 and - 7, -5+( - 7)=-12. We can conjecture that the sum of two negative integers is a negative integer.

Step2: Deductive reasoning to prove

Let \(a\) and \(b\) be negative integers. So \(a=-m\) where \(m>0\) and \(b = - n\) where \(n>0\). Then \(a + b=-m+( - n)=-(m + n)\). Since \(m>0\) and \(n>0\), \(m + n>0\), so \(-(m + n)<0\), which means \(a + b\) is a negative integer.

26.

Step1: Distribute -3

We have \(-3(6x - 1)=-18x+3\), so the equation becomes \(-18x + 3=6x-9\) (Distributive property \(a(b - c)=ab - ac\)).

Step2: Add 18x to both sides

\(-18x+18x + 3=6x+18x-9\), which simplifies to \(3 = 24x-9\) (Addition - property of equality: if \(a=b\), then \(a + c=b + c\)).

Step3: Add 9 to both sides

\(3+9=24x-9 + 9\), so \(12 = 24x\) (Addition - property of equality).

Step4: Divide both sides by 24

\(\frac{12}{24}=\frac{24x}{24}\), and \(x=\frac{1}{2}\) (Division - property of equality: if \(a=b\) and \(c
eq0\), then \(\frac{a}{c}=\frac{b}{c}\)).

27.

The statement \(\overline{JK}=\overline{JK}\) illustrates the Reflexive property of congruence for line - segments. Any line - segment is congruent to itself.

Answer:

Reflexive property of congruence for line - segments

28.