Which operator can be inserted to make the statement true?(2 points)

Question

8−18⋅12−1

9+4−11⋅2

Answer 1

To determine which operator can be inserted to make the statements true, we need to evaluate both expressions separately using potential operators.

For the first expression:

8−18⋅12−1

Let's analyze it without an operator first:

1. Calculate \( 18 \cdot 12 = 216 \)
2. Then, \( 8 - 216 - 1 = 8 - 216 - 1 = 8 - 217 = -209 \)

This expression doesn't create a true statement as it stands. To make it true, we could think of inserting a positive operator that gives a different arithmetic outcome.

Next, let’s analyze the second expression:

9 + 4 − 11 ⋅ 2

Calculate step-by-step:

1. Calculate \( 11 \cdot 2 = 22 \)
2. Now calculate: \( 9 + 4 - 22 = 13 - 22 = -9 \)

This is also false as it stands.

The objective is to find a common operator or something to balance both expressions such that they yield equal outputs or a true statement.

However, the primary function is to show the relationships among operations on numbers.

If you simply want to balance these two statements to be equal or true, you'd have to check possible operators and their locations such as +, -, *, or / that could lead to a valid statement.

Without the specific operator, it's difficult to identify any other operators that would apply here to reach truth since these statements could have numerous possibilities to ascertain depending on the values involved.

To finalize: After working through the calculations, it looks like I initially misled you. Unfortunately, the task requires more specificity. Would you like me to suggest common relationships between values instead? Or, could you specify if you're looking for the results or true equations?

Answer 2

Which set expresses the solution to the compound inequality −3≤2x−5<1
?(2 points)
Responses

{x | 2≤x<6}
{x | 2≤x<6}

{x:3.5≤x<5.5}
{x:3.5≤x<5.5}

{x | −1.5≤x<0.5}
{x | −1.5≤x<0.5}

{x:1≤x<3}

Answer 3

To solve the compound inequality \(-3 \leq 2x - 5 < 1\), we can break it into two separate inequalities and solve each part.

1. First part: \(-3 \leq 2x - 5\)

   To isolate \(x\), first add 5 to both sides: \[ -3 + 5 \leq 2x \] \[ 2 \leq 2x \] Now, divide both sides by 2: \[ 1 \leq x \] or \[ x \geq 1 \]

2. Second part: \(2x - 5 < 1\)

   Again, isolate \(x\) by adding 5 to both sides: \[ 2x < 1 + 5 \] \[ 2x < 6 \] Now, divide both sides by 2: \[ x < 3 \]

Now, we can combine the results from both inequalities:

\[ 1 \leq x < 3 \]

In interval notation, this is expressed as:

\({ x \mid 1 \leq x < 3 }\)

So out of the provided response options, the correct set expressing the solution to the compound inequality \(-3 \leq 2x - 5 < 1\) is:

\({x: 1 \leq x < 3}\)

Answer 4

What property of equality is shown in this step of solving the equation?



6m+12=5m+7
6m+12−12=5m+7−12
(2 points)
Responses

Addition Property of Equality
Addition Property of Equality

Division Property of Equality
Division Property of Equality

Subtraction Property of Equality
Subtraction Property of Equality

Multiplication Property of Equality

Answer 5

In the given steps of solving the equation:

1. Start with the equation: \(6m + 12 = 5m + 7\)
2. Then subtract 12 from both sides: \(6m + 12 - 12 = 5m + 7 - 12\)

When you subtract 12 from both sides of the equation, you are using the Subtraction Property of Equality, which states that if you subtract the same number from both sides of an equation, the two sides remain equal.

Therefore, the correct response is:

Subtraction Property of Equality

Answer 6

Find the solution set of the following equation:

(Use set notation)

(2 points)
d3−6+2d=3(d−4)

Answer 7

To solve the equation \(d^3 - 6 + 2d = 3(d - 4)\), we start by simplifying it step by step.

1. Distribute on the right side: \[ 3(d - 4) = 3d - 12 \]

2. Rewriting the equation: \[ d^3 - 6 + 2d = 3d - 12 \]

3. Rearranging everything to one side: \[ d^3 - 6 + 2d - 3d + 12 = 0 \] Combine like terms: \[ d^3 - d + 6 = 0 \]

Now we need to solve the cubic equation \(d^3 - d + 6 = 0\).

4. Finding rational roots: We can use the Rational Root Theorem to test possible rational roots, which could be factors of \(6\). Testing \(d = -2\):

   \[ (-2)^3 - (-2) + 6 = -8 + 2 + 6 = 0 \]

   Thus, \(d = -2\) is a root. Now, we can factor \(d + 2\) out of the polynomial \(d^3 - d + 6\).

5. Factoring the polynomial: Using synthetic division to divide \(d^3 - d + 6\) by \(d + 2\):

   • Bring down the leading coefficient to start: -1 (for \(d^3\)), 0 (for \(2d^2\)), -1 (for \(-d\)), and 6 (the constant term).

\[ \begin{array}{r|rrrr} -2 & 1 & 0 & -1 & 6 \ & & -2 & 4 & -6 \ \hline & 1 & -2 & 3 & 0 \ \end{array} \]

The result from synthetic division is \(d^2 - 2d + 3\).

6. Write the factorization: \[ d^3 - d + 6 = (d + 2)(d^2 - 2d + 3) \]

7. Solve \(d^2 - 2d + 3 = 0\) using the quadratic formula: The quadratic formula is given by: \[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = -2\), and \(c = 3\).

   \[ d = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} \] Simplifying the expression: \[ d = \frac{2 \pm \sqrt{4 - 12}}{2} \] \[ d = \frac{2 \pm \sqrt{-8}}{2} \] \[ d = \frac{2 \pm 2i\sqrt{2}}{2} \] This simplifies to: \[ d = 1 \pm i\sqrt{2} \]

So, the three solutions to the original equation are:

1. \(d = -2\)

2. \(d = 1 + i\sqrt{2}\)

3. \(d = 1 - i\sqrt{2}\)

4. Solution Set in Set Notation:

The solution set in set notation is:

\[ { -2, 1 + i\sqrt{2}, 1 - i\sqrt{2} } \]

Answer 8

Find the solution to the inequality −11g+18<−5g+54
(2 points)
Responses

g<−6
g<−6

g<−92
g<−92

g>−92
g is greater than negative 9 halves

g>−6
g is greater than negative 6

Answer 9

To solve the inequality \(-11g + 18 < -5g + 54\), we will follow these steps:

1. Isolate the variable terms on one side: Start by adding \(11g\) to both sides: \[ 18 < 6g + 54 \]

2. Subtract \(54\) from both sides: \[ 18 - 54 < 6g \] \[ -36 < 6g \]

3. Divide both sides by \(6\): \[ \frac{-36}{6} < g \] \[ -6 < g \] or, equivalently, \[ g > -6 \]

So, the solution to the inequality is:

g > -6

Thus, the correct response is:

g > -6 (g is greater than negative 6)

Answer 10

Find the solution set of 4x−3>13
and x+8>10


(Use set notation)

(2 points)

Answer 11

To find the solution set for the compound inequalities \(4x - 3 > 13\) and \(x + 8 > 10\), we will solve each inequality separately.

Solve the first inequality:

1. Starting with \(4x - 3 > 13\):
   • Add 3 to both sides: \[ 4x > 16 \]
   • Divide both sides by 4: \[ x > 4 \]

Solve the second inequality:

2. Starting with \(x + 8 > 10\):
   • Subtract 8 from both sides: \[ x > 2 \]

Combine the two results:

Now we have:

• \(x > 4\) from the first inequality
• \(x > 2\) from the second inequality

The more restrictive condition is \(x > 4\), because any \(x\) that satisfies \(x > 4\) will also satisfy \(x > 2\).

Solution Set:

In set notation, the solution set is: \[ { x \mid x > 4 } \]

This indicates that \(x\) can be any value greater than 4.