Question

13 multiple choice 4 points
complete the following arithmetic sequence.
□, 6, □, □, 19
1 1/2; 10 1/2; 14 2/3
1 2/3; 10 1/3; 14 2/3
2; 10; 14
1 3/4; 10 1/2; 14 3/4
14 multiple choice 4 points
which of the following is the graph of |x| > -2?

Explanation:

Step1: Recall arithmetic - sequence formula

The formula for an arithmetic sequence is $a_n=a_1+(n - 1)d$, where $a_n$ is the $n$th term, $a_1$ is the first - term, $n$ is the term number, and $d$ is the common difference. Here, $a_2 = 6$ and $a_5=19$.

Step2: Find the common difference $d$

Using the formula $a_n=a_1+(n - 1)d$, for $a_2=a_1 + d$ and $a_5=a_1+4d$. We know that $a_2 = 6$ so $a_1 + d=6$, and $a_5 = 19$ so $a_1+4d=19$. Subtract the first equation from the second: $(a_1 + 4d)-(a_1 + d)=19 - 6$. This simplifies to $3d=13$, so $d=\frac{13}{3}$.

Step3: Find the first term $a_1$

Since $a_1 + d=6$ and $d=\frac{13}{3}$, then $a_1=6 - d=6-\frac{13}{3}=\frac{18 - 13}{3}=\frac{5}{3}=1\frac{2}{3}$.

Step4: Find the third and fourth terms

The third term $a_3=a_1 + 2d=\frac{5}{3}+2\times\frac{13}{3}=\frac{5 + 26}{3}=\frac{31}{3}=10\frac{1}{3}$. The fourth term $a_4=a_1+3d=\frac{5}{3}+3\times\frac{13}{3}=\frac{5 + 39}{3}=\frac{44}{3}=14\frac{2}{3}$.

Step1: Analyze the absolute - value inequality $|x|>-2$

The absolute - value of any real number $x$, i.e., $|x|$, is always non - negative. That is, $|x|\geq0$ for all real numbers $x$. Since $0>-2$, the inequality $|x|>-2$ is true for all real numbers $x$.

Step2: Determine the graph

The graph of the set of all real numbers on a number line is a continuous line that extends infinitely in both the positive and negative directions.

Answer:

1. $1\frac{2}{3};10\frac{1}{3};14\frac{2}{3}$
2. The first graph (the one that shows a continuous line extending from negative infinity to positive infinity on the number line)