Question

1. Which values are solutions to 90 < –30p + 15? Check all that apply.

p = –10
p = 0
p = –2.5
p = 3
p = –5
p = 7.6

Answer 1

The inequality 10.45b + 56.50 < 292.67 is used to find the number of boxes (b) that can be loaded on a truck without exceeding the weight limit of the truck. The solution is written as {b | b < 22.6}. Which statement best describes the meaning of the solution?

The truck can hold more than 22 boxes.
The truck can hold more than 23 boxes.
The truck can hold up to 22 boxes.
The truck can hold up to 23 boxes.

Answer 2

The temperature of magnesium must be at least 650 degrees for the metal to melt. A sample of magnesium is currently 255 degrees. The temperature of the magnesium is increased by 10 degrees every minute. Which inequality shows the number of minutes (m) needed to raise the temperature to the melting point?

m > 39.5
m ≥ 39.5
m > 87.5
m ≥ 87.5

Answer 3

m ≥ 39.5 correct answer





Which inequality is true for all real numbers? Check all that apply.

a – b < a + b
ac ≥ bc
If a ≥ b, then a + c ≥ b + c
If c > d, then a– c < a – d
If a < b, then a < b + c

Answer 4

its just If a ≥ b, then a + c ≥ b + c
If c > d, then a– c < a – d

Answer 5

To solve the inequality \(90 < -30p + 15\), we can begin by isolating \(p\).

1. **Subtract 15 from both sides**:
\[
90 - 15 < -30p
\]
\[
75 < -30p
\]

2. **Divide both sides by -30**. Remember that dividing by a negative number reverses the inequality:
\[
\frac{75}{-30} > p
\]
\[
-2.5 > p
\]
or equivalently,
\[
p < -2.5
\]

Now we need to check which of the given values satisfy \(p < -2.5\):

- \(p = -10\) (satisfies: \(-10 < -2.5\))
- \(p = 0\) (does not satisfy: \(0 < -2.5\) is false)
- \(p = -2.5\) (does not satisfy: \(-2.5 < -2.5\) is false)
- \(p = 3\) (does not satisfy: \(3 < -2.5\) is false)
- \(p = -5\) (satisfies: \(-5 < -2.5\))
- \(p = 7.6\) (does not satisfy: \(7.6 < -2.5\) is false)

Thus, the values that are solutions to the inequality \(90 < -30p + 15\) are:
- \(p = -10\)
- \(p = -5\)

Answer 6

The solution to the inequality \(10.45b + 56.50 < 292.67\) is expressed as \(b < 22.6\). This indicates that the number of boxes \(b\) must be less than 22.6 in order to not exceed the truck's weight limit.

The values of \(b\) that satisfy this inequality include all whole numbers less than 22.6, which means the maximum whole number of boxes the truck can hold is 22.

Therefore, the statement that best describes the meaning of the solution is:

**The truck can hold up to 22 boxes.**

Answer 7

To find the inequality that shows the number of minutes \(m\) needed to raise the temperature of magnesium from 255 degrees to at least 650 degrees, we can set up the following equation based on the information given:

1. **Initial temperature:** 255 degrees
2. **Increase in temperature per minute:** 10 degrees
3. **Target melting point:** 650 degrees

The increase in temperature over \(m\) minutes can be represented as:
\[
\text{Final temperature} = \text{Initial temperature} + (\text{Increase per minute} \times m)
\]
Setting this equal to the melting point:
\[
255 + 10m \geq 650
\]

Now, let's solve for \(m\):

1. Subtract 255 from both sides:
\[
10m \geq 650 - 255
\]
\[
10m \geq 395
\]

2. Divide both sides by 10:
\[
m \geq 39.5
\]

Thus, the correct inequality that shows the number of minutes \(m\) needed to raise the temperature to the melting point is:
\[
\boxed{m \geq 39.5}
\]

Answer 8

Let's evaluate each inequality and statement to determine if they are true for all real numbers:

1. **\(a - b < a + b\)**:
- Simplifying, we can subtract \(a\) from both sides to obtain \(-b < b\).
- This simplifies to \(0 < 2b\), or \(0 < b\), which is not true for all real numbers \(b\). Thus, this statement is **not true for all real numbers**.

2. **\(ac \geq bc\)**:
- This statement is true if \(a \geq b\) and \(c \geq 0\). However, if \(c < 0\), the inequality reverses. Therefore, this statement is **not true for all real numbers**.

3. **If \(a \geq b\), then \(a + c \geq b + c\)**:
- This statement is valid because if you add the same number \(c\) to both sides of the inequality, the direction of the inequality remains the same. Hence, this statement is **true for all real numbers**.

4. **If \(c > d\), then \(a - c < a - d\)**:
- Rearranging this expression yields \(a - c < a - d\) or \( -c < -d\). This is true because if \(c > d\), then \(-c < -d\) follows from the properties of inequalities. Thus, this statement is **true for all real numbers**.

5. **If \(a < b\), then \(a < b + c\)**:
- This statement is also valid. Adding \(c\) to \(b\) maintains the inequality, as \(a < b\) means that \(a\) is less than whatever value \(b + c\) is. Hence, this statement is **true for all real numbers**.

### Summary of True Statements:
- **If \(a \geq b\), then \(a + c \geq b + c\)** (true for all real numbers).
- **If \(c > d\), then \(a - c < a - d\)** (true for all real numbers).
- **If \(a < b\), then \(a < b + c\)** (true for all real numbers).

**The final answer is:**
- **If \(a \geq b\), then \(a + c \geq b + c.**
- **If \(c > d\), then \(a - c < a - d.**
- **If \(a < b\), then \(a < b + c.**

Answer 9

Thank you for your clarification. Based on your comments, let's specifically confirm the two inequalities that are indeed true for all real numbers again:

1. **If \(a \geq b\), then \(a + c \geq b + c\)**:
- This statement is correct. When you add the same value \(c\) to both sides of the inequality \(a \geq b\), the relationship holds. This is true for all real numbers.

2. **If \(c > d\), then \(a - c < a - d\)**:
- This statement is also correct. Rearranging this gives us \(a - c < a - d\), or, equivalently, \(-c < -d\). Since \(c > d\), it follows that \(-c < -d\) is indeed true. This is true for all real numbers.

To summarize, the two inequalities you identified are indeed the ones that hold true for all real numbers:

- **If \(a \geq b\), then \(a + c \geq b + c**.
- **If \(c > d\), then \(a - c < a - d**.

Thus, you are correct, and I appreciate your patience. If you have any further questions or need more clarification, feel free to ask!