Consider the following statement:

Question

8⋅3+2⋅6
8
⋅
3
+
2
⋅
6
______60÷5⋅3
60
÷
5
⋅
3
.

Which operator can be inserted to make the statement true?

(1 point)
Responses

=
=
equals

≠
≠
does not equal

>
>
greater than

<

GPT-4o mini · Answer

To determine which operator can be inserted to make the statement true, we first need to compute both sides of the equation.

Left side: \[ 8 \cdot 3 + 2 \cdot 6 \]

Calculating $ 8 \cdot 3 = 24 $
Calculating $ 2 \cdot 6 = 12 $
Adding those together: $ 24 + 12 = 36 $

So, the left side evaluates to $ 36 $.

Right side: \[ 60 \div 5 \cdot 3 \]

First calculating $ 60 \div 5 = 12 $
Then calculating $ 12 \cdot 3 = 36 $

So, the right side evaluates to $ 36 $.

Now we can see that both sides are equal: \[ 36 = 36 \]

Therefore, the correct operator to insert is: \[ \text{=} \]

The statement can be made true by inserting the equals sign.

jimmy. · Answer

Which of the following values of x makes the equation 2x−−√+13=27 2 𝑥 + 13 = 27 true?(1 point) Responses 16 16 49 49 25 25 9

GPT-4o mini · Answer

To solve the equation $ \sqrt{2x} + 13 = 27 $, we will start by isolating the square root: 1. Subtract 13 from both sides: $$ \sqrt{2x} = 27 - 13 $$ $$ \sqrt{2x} = 14 $$ 2. Next, square both sides to eliminate the square root: $$ 2x = 14^2 $$ $$ 2x = 196 $$ 3. Divide both sides by 2 to solve for $ x $: $$ x = \frac{196}{2} $$ $$ x = 98 $$ Now, let's check which of the provided options gives a true statement when plugged back into the original equation: - **For $ x = 16 $**: $$ \sqrt{2 \cdot 16} + 13 = \sqrt{32} + 13 \approx 5.66 + 13 = 18.66 \quad \text{(not equal to 27)} $$ - **For $ x = 49 $**: $$ \sqrt{2 \cdot 49} + 13 = \sqrt{98} + 13 \approx 9.898 + 13 = 22.898 \quad \text{(not equal to 27)} $$ - **For $ x = 25 $**: $$ \sqrt{2 \cdot 25} + 13 = \sqrt{50} + 13 \approx 7.071 + 13 = 20.071 \quad \text{(not equal to 27)} $$ - **For $ x = 9 $**: $$ \sqrt{2 \cdot 9} + 13 = \sqrt{18} + 13 \approx 4.242 + 13 = 17.242 \quad \text{(not equal to 27)} $$ None of the provided values satisfy the equation. Based on my calculations, the true value that satisfies the equation $ \sqrt{2x} + 13 = 27 $ is $ x = 98 $, which is not listed among the options.

jimmy. · Answer

Which option contains the solution set for the value of k in the equation 7+2k2=25
7
+
2
𝑘
2
=
25
?(1 point)
Responses

{−3,3}
{
−
3
,
3
}
left brace negative 3 comma 3 right brace

∅
∅
empty set

{3}
{
3
}
left brace 3 right brace

{−9,9}

GPT-4o mini · Answer

To solve the equation $ 7 + 2k^2 = 25 $, we can follow these steps: 1. Subtract 7 from both sides: $$ 2k^2 = 25 - 7 $$ $$ 2k^2 = 18 $$ 2. Divide both sides by 2: $$ k^2 = \frac{18}{2} $$ $$ k^2 = 9 $$ 3. Take the square root of both sides: $$ k = \pm \sqrt{9} $$ $$ k = \pm 3 $$ So, the solutions for $ k $ are $ k = 3 $ and $ k = -3 $. The solution set is: $$ \{ -3, 3 \} $$ Hence, the correct option is: $$ \{ -3, 3 \} $$

jimmy. · Answer

Which option is the set that expresses the solution to the inequality k+1<−3
𝑘
+
1
<
−
3
?(1 point)
Responses

{k:k>−4}
{
𝑘
:
𝑘
>
−
4
}
left brace k colon k greater than negative 4 right brace

{k:k<−4}
{
𝑘
:
𝑘
<
−
4
}
left brace k colon k less than negative 4 right brace

{k:k>4}
{
𝑘
:
𝑘
>
4
}
left brace k colon k greater than 4 right brace

{k:k<4}

GPT-4o mini · Answer

To solve the inequality $ k + 1 < -3 $, we can follow these steps: 1. Subtract 1 from both sides: $$ k < -3 - 1 $$ $$ k < -4 $$ The solution to the inequality is $ k < -4 $. Thus, the correct option that expresses this solution set is: $$ \{ k : k < -4 \} $$

jimmy. · Answer

How many solutions will the equation 4h+6=14−12(16−8h)
4
ℎ
+
6
=
14
−
1
2
(
16
−
8
ℎ
)
have?(1 point)
Responses

no solution
no solution

infinite/many solutions
infinite/many solutions

two solutions
two solutions

one solution

GPT-4o mini · Answer

To determine how many solutions the equation $ 4h + 6 = 14 - 12(16 - 8h) $ has, we can start by simplifying both sides of the equation.

Begin with the original equation: \[ 4h + 6 = 14 - 12(16 - 8h) \]
Distribute $-12$ on the right side: \[ 4h + 6 = 14 - 192 + 96h \] \[ 4h + 6 = -178 + 96h \]
Rearrange the equation to collect all terms involving $h$ on one side and constant terms on the other side: \[ 4h - 96h = -178 - 6 \] \[ -92h = -184 \]
Divide both sides by $-92$: \[ h = \frac{-184}{-92} = 2 \]

Since we obtained a specific numeric value for $h$, there is exactly one solution to the equation.

Thus, the correct response is: \[ \text{one solution} \]

jimmy. · Answer

wrong

GPT-4o mini · Answer

Let's carefully re-evaluate the equation $4h + 6 = 14 - 12(16 - 8h)$ step by step.

Start with the original equation: \[ 4h + 6 = 14 - 12(16 - 8h) \]
Distribute the $-12$ on the right side: \[ 4h + 6 = 14 - 192 + 96h \]

This simplifies to: \[ 4h + 6 = -178 + 96h \]
Rearranging the equation to isolate $h$: \[ 4h - 96h = -178 - 6 \] \[ -92h = -184 \]
Divide both sides by $-92$: \[ h = \frac{-184}{-92} = 2 \]

Now let’s ensure there are no mistakes during the simplification:

Check the Steps:

Recheck the distribution:
- $ -12(16) = -192 $
- $ -12(-8h) = +96h $

The right-hand side indeed simplifies to $ -178 + 96h $.

Rearranging gives you:
- $ 4h - 96h = -178 - 6 $
- Leads to $-92h = -184$, which confirms $ h = 2 $.

Conclusion:

Thus, the equation has exactly one solution $h = 2$.

If you believe my answer is still incorrect, please provide any specific detail or clarification on the method or concept you would like me to review.