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Which set of numbers is included as part of the solution set of the compound inequality

x < 6 or x > 10?

{–7, –1.7, 6.1, 10}
{–3, 4.5, 13.6, 19}
{0, 6, 9.8, 14}
{8.5, 9.1}

11 answers

To find the set of numbers that is included as part of the solution set of the compound inequality \(x < 6\) or \(x > 10\), we need to identify which numbers from the given sets are either less than 6 or greater than 10.

Let's analyze each set:

  1. Set: {–7, –1.7, 6.1, 10}

    • –7 is less than 6 (included).
    • –1.7 is less than 6 (included).
    • 6.1 is greater than 6 but not greater than 10 (not included).
    • 10 is not less than 6 and not greater than 10 (not included).
    • Included numbers: –7, –1.7

  2. Set: {–3, 4.5, 13.6, 19}

    • –3 is less than 6 (included).
    • 4.5 is less than 6 (included).
    • 13.6 is greater than 10 (included).
    • 19 is greater than 10 (included).
    • Included numbers: –3, 4.5, 13.6, 19

  3. Set: {0, 6, 9.8, 14}

    • 0 is less than 6 (included).
    • 6 is not less than 6 (not included).
    • 9.8 is less than 10 (not included).
    • 14 is greater than 10 (included).
    • Included numbers: 0, 14

  4. Set: {8.5, 9.1}

    • 8.5 is less than 10 (not included).
    • 9.1 is less than 10 (not included).
    • No included numbers.

Now, we summarize which sets include numbers from the solution set of the compound inequality \(x < 6\) or \(x > 10\):

  • Set 1: {–7, –1.7} (numbers less than 6)
  • Set 2: {–3, 4.5, 13.6, 19} (numbers less than 6 and greater than 10)
  • Set 3: {0, 14} (numbers less than 6 and greater than 10)
  • Set 4: {8.5, 9.1} (no included numbers)

The only set from the options that has numbers in the solution set is:

{–3, 4.5, 13.6, 19}

The compound inequality 8.00 ≤ x < 9.50 represents all values, x, for which college students are paid hourly as teacher assistants. What is another way of writing this compound inequality?

x ≤ 8.00 and x < 9.50
x ≤ 8.00 or x < 9.50
x ≥ 8.00 and x < 9.50
x ≥ 8.00 or x < 9.50

The compound inequality \(8.00 \leq x < 9.50\) means that \(x\) can take any value greater than or equal to \(8.00\) and less than \(9.50\).

This can be rewritten in another way by expressing the conditions clearly:

  • \(x\) must be greater than or equal to 8.00
  • \(x\) must be less than 9.50

Thus, the correct way to restate this inequality is:

\(x \geq 8.00 \text{ and } x < 9.50\)

Therefore, the correct answer is:

x ≥ 8.00 and x < 9.50

In order for a gear to work in a piece of machinery, the radius of the gear, r, must be greater than 4 cm, but not exceed 4.1 cm. Which compound inequality represents the situation?

r > 4 and r ≤ 4.1
r > 4 or r ≤ 4.1
r < 4 and r ≥ 4.1
r < 4 or r ≥ 4.1

To express the conditions for the radius of the gear, we need to translate the requirements given:

  1. The radius of the gear, \(r\), must be greater than 4 cm: this can be written as \(r > 4\).
  2. The radius must also not exceed 4.1 cm: this can be written as \(r \leq 4.1\).

Putting these two conditions together forms a compound inequality:

\[ r > 4 \quad \text{and} \quad r \leq 4.1 \]

Thus, the correct compound inequality that represents the situation is:

r > 4 and r ≤ 4.1

The image shows a number line with two highlighted sections. The first section is highlighted from the start of the number line up to but not including 5, indicated by an open circle at 5. The second section is highlighted from 9, indicated by a closed circle, extending to the right without end.

The question asks which compound inequality is represented by the graph. The correct answer is: ( x < 5 ) or ( x \geq 9 ).

Based on your description of the number line:

  1. The first highlighted section is from the start of the number line up to but not including 5, indicated by an open circle at 5. This means it represents the inequality \(x < 5\).

  2. The second highlighted section starts at 9, indicated by a closed circle, and extends to the right without end. This represents the inequality \(x \geq 9\).

Since the two highlighted sections do not overlap and are independent of each other, we combine them with "or."

Therefore, the compound inequality represented by the graph is:

\(x < 5 \text{ or } x \geq 9\)

Your provided answer is indeed correct.

it is 5 < x ≤ 9

The image shows a question above four number line graphs. The question asks, “Which graph represents the inequality ( x \leq -2 ) or ( x \geq 0 )?”

First Graph: Shading to the left from -2 with an open circle on -2. This represents ( x < -2 ).
Second Graph: Shading to the right from -2 with a closed circle on -2. This represents ( x \geq -2 ).
Third Graph: Two shaded regions: one to the left from 0 with an open circle on 0, and one to the right from 0 with a closed circle on 0. This represents ( x \neq 0 ).
Fourth Graph: Two shaded regions: one to the left from -2 with a closed circle on -2, and one to the right from 0 with a closed circle on 0. This represents ( x \leq -2 ) or ( x \geq 0 ).

Based on your description of the graphs:

  1. First Graph: Shading to the left from -2 with an open circle on -2. This represents \(x < -2\).
  2. Second Graph: Shading to the right from -2 with a closed circle on -2. This represents \(x \geq -2\).
  3. Third Graph: Two shaded regions, one to the left from 0 with an open circle on 0, and one to the right from 0 with a closed circle on 0. This represents a condition that indicates \(x \neq 0\) (which isn't the correct representation for the given inequality).
  4. Fourth Graph: Two shaded regions, one to the left from -2 with a closed circle on -2, and one to the right from 0 with a closed circle on 0. This represents \(x \leq -2\) or \(x \geq 0\).

Since the question asks for the graph that represents the inequality \( (x \leq -2) \text{ or } (x \geq 0) \), the fourth graph is indeed the correct choice because it shows the correct shading and includes closed circles at the endpoints.

Therefore, the answer is that the fourth graph correctly represents the inequality \( (x \leq -2) \text{ or } (x \geq 0) \).

pick something else other than 4

Let's analyze the graphs again based on the description you've provided, as you requested to select something other than the fourth graph.

  1. First Graph: Shading to the left from -2 with an open circle on -2. This represents \(x < -2\).

  2. Second Graph: Shading to the right from -2 with a closed circle on -2. This represents \(x \geq -2\).

  3. Third Graph: Two shaded regions – one to the left from 0 with an open circle on 0, and one to the right from 0 with a closed circle on 0. This represents \(x \neq 0\), but it doesn't accurately reflect \(x \leq -2\) or \(x \geq 0\).

  4. Fourth Graph: Two shaded regions – one to the left from -2 with a closed circle on -2, and one to the right from 0 with a closed circle on 0. This is the correct representation of \( (x \leq -2) \text{ or } (x \geq 0) \).

Given that the fourth graph is indeed the one that accurately represents your inequality and you've asked for a choice other than that, none of the other graphs accurately represent \( (x \leq -2) \text{ or } (x \geq 0) \).

However, if I must choose one of the other graphs, I would pick the second graph which represents \(x \geq -2\). While it does not fully depict the original inequality, it at least represents a region that includes values greater than or equal to -2.

Thus, if you are looking for a different selection, I would suggest:

  • Second Graph: \(x \geq -2\).

Please keep in mind that this graph does not represent the required inequality completely, but it’s the next closest option.