Question

find the solution set for each replacement set where x ≤ 4.

1. r = {0, 1, 2, 3, 4, ...}

s = {0, 1, 2, 3, 4}

1. r = {0, 2, 4, 6, 8, ...}
2. r = {1, 3, 5, 7, 9, ...}
3. r = {1, 10, 100, 1000, ...}
4. r = {1, 4, 8, 12, 16, ...}
5. r = {6, 8, 10, 12, 14, ...}

write an inequality to describe the graph.

1. ←—•———→

-2 -1 0 1 2 3
x ≥ -1

1. ←—•———→

-3 -2 -1 0 1 2

1. ←—•———→

0 1 2 3 4 5

1. ←—⊕———→

-3 -2 -1 0 1 2

1. ←—⊕———→

-14 -13 -12 -11

1. ←—•———→

5 6 7 8 9 10
graph each inequality on a number line.

1. x ≥ 5
2. x ≤ 4
3. -15 > x
4. -2 > x
5. x > -3
6. x ≥ 100

Explanation:

Step1: Identify the problem type

We have multiple sub - problems related to inequalities and solution sets.

Step2: For finding solution sets

For example, in problem 15, the replacement set $R=\{0,1,2,3,4,\ldots\}$ and we need values of $x$ such that $x\leq4$. The solution set $S = \{0,1,2,3,4\}$ as these are the elements from $R$ that satisfy the inequality.

Step3: For writing inequalities from graphs

In problem 21, the graph has a closed - circle at $x = - 1$ and the line goes to the right. So the inequality is $x\geq - 1$.

Step4: For graphing inequalities

For $x\geq5$ (problem 27), we draw a closed - circle at 5 on the number line and draw a line to the right. For $-15>x$ (problem 29), we draw an open - circle at $-15$ and draw a line to the left.

Answer:

1. $S=\{0,1,2,3,4\}$
2. For $R = \{0,2,4,6,8,\ldots\}$ and $x\leq4$, $S=\{0,2,4\}$
3. For $R=\{1,3,5,7,9,\ldots\}$ and $x\leq4$, $S = \{1,3\}$
4. For $R=\{1,10,100,1000,\ldots\}$ and $x\leq4$, $S=\{1\}$
5. For $R=\{1,4,8,12,16,\ldots\}$ and $x\leq4$, $S=\{1,4\}$
6. For $R=\{6,8,10,12,14,\ldots\}$ and $x\leq4$, $S=\varnothing$
7. $x\geq - 1$
8. The graph has a closed - circle at 1 and goes to the right, so $x\geq1$
9. The graph has an open - circle at 4 and goes to the right, so $x>4$
10. The graph has an open - circle at $-1$ and goes to the right, so $x>-1$
11. The graph has an open - circle at $-13$ and goes to the right, so $x>-13$
12. The graph has a closed - circle at 7 and goes to the right, so $x\geq7$
13. Graph: Draw a closed - circle at 5 on the number line and a line to the right.
14. Graph: Draw a closed - circle at 4 on the number line and a line to the left.
15. Graph: Draw an open - circle at $-15$ on the number line and a line to the left.
16. Graph: Draw an open - circle at $-2$ on the number line and a line to the left.
17. Graph: Draw an open - circle at $-3$ on the number line and a line to the right.
18. Graph: Draw a closed - circle at 100 on the number line and a line to the right.