Choose the compound inequality that represents each phrase.
All real numbers that are less than –3 or greater than or equal to 5
(1 point)
Responses
x < –3 or x ≥ 5
x < –3 or x ≥ 5
x < 5 or x ≥ –3
x < 5 or x ≥ –3
x ≤ –3 or x > 5
x ≤ –3 or x > 5
x ≤ –3 or x ≤ 5
The phrase "All real numbers that are less than –3 or greater than or equal to 5" would be best represented by the compound inequality:
x < –3 or x ≥ 5
So the correct response is:
x < –3 or x ≥ 5
The time a cake must bake is between 25 minutes and 30 minutes, inclusive.
(1 point)
Responses
25 < x < 30
25 < x < 30
25 ≤ x ≤ 30
25 ≤ x ≤ 30
25 ≤ x ≥ 30
25 ≤ x ≥ 30
25 < x > 30
The phrase "The time a cake must bake is between 25 minutes and 30 minutes, inclusive" means that the baking time can be exactly 25 minutes, exactly 30 minutes, or any time in between. So, the correct way to represent this would be to use a compound inequality that includes both endpoints. Therefore, the correct response is:
25 ≤ x ≤ 30
So you would choose the following response:
25 ≤ x ≤ 30
Solve. 5 < x – 2 < 11
(1 point)
Responses
5 < x < 11
5 < x < 11
7 < x < 13
7 < x < 13
7 < x < 11
7 < x < 11
5 < x < 16
5 < x < 16
To solve the compound inequality 5 < x – 2 < 11, you need to isolate the variable x in the middle. You do this by adding 2 to all three parts of the inequality:
5 + 2 < x – 2 + 2 < 11 + 2
Which simplifies to:
7 < x < 13
Therefore, the correct response is:
7 < x < 13
Which graph shows the solution of 9 – c < 2 or –3c > 15?
(1 point)
Responses
–10 to 10 number line
Image with alt text: –10 to 10 number line
–10 to 10 number lineA closed point is shown at negative 5. A line goes from negative 5 to the arrow at the left end of the line. Another closed point is shown at 7. A line goes from 7 to the arrow at the right end of the number line.
Image with alt text: –10 to 10 number line A closed point is shown at negative 5. A line goes from negative 5 to the arrow at the left end of the line. Another closed point is shown at 7. A line goes from 7 to the arrow at the right end of the number line.
–10 to 10 number lineAn open point is shown at negative 5. A line goes from negative 5 through negative 6, negative 7, negative 8, negative 9, and negative 10 to the arrow at the left end of the line. Another open point is shown at 7. A line goes from 7 through 8, 9, and 10 to the arrow at the right end of the number line.
Image with alt text: –10 to 10 number line An open point is shown at negative 5. A line goes from negative 5 through negative 6, negative 7, negative 8, negative 9, and negative 10 to the arrow at the left end of the line. Another open point is shown at 7. A line goes from 7 through 8, 9, and 10 to the arrow at the right end of the number line.
–10 to 10 number line
First, we'll solve each part of the compound inequality individually:
For the inequality 9 – c < 2, we'll subtract 9 from both sides to isolate c:
- c < 2 - 9
- c < -7
To find c, we have to divide by -1 (and remember that dividing by a negative number reverses the inequality sign):
c > 7
For the inequality –3c > 15, we'll divide both sides by -3 (again, remembering to reverse the inequality sign when dividing by a negative number):
c < -5
So the two parts of the compound inequality are:
c > 7 or c < -5
The graph representing these solutions should have a line extending to the right from 7 with an open circle at c = 7 (because it is not "greater than or equal to") and a line extending to the left from -5 with an open circle at c = -5 (for the same reason).
The correct graph would therefore show:
- An open circle at -5 with a line extending to the left towards negative infinity.
- An open circle at 7 with a line extending to the right towards positive infinity.
Unfortunately, without the ability to see the images provided with the responses, I cannot tell you which graphic representation matches this description. However, based on the text descriptions, it seems that a graph should resemble the following character-based representation (if we assume a one-dimensional number line):
`<---( -5 )------------------------( 7 )--->`
The parentheses represent open circles at -5 and 7, indicating that those values are not included in the solution set, and the arrows indicate that the line continues indefinitely in both directions. Please select the image that corresponds to this description. If any of the text descriptions provided accurately matches this, that would be the correct choice.