Solve for xx and graph the solution on the number line below.

Question

minus, 2, x, plus, 1, is less than, minus, 9, and, minus, 2, x, plus, 1, is greater than or equal to, minus, 13
−2x+1<−9and
−2x+1≥−13
Answer
Attempt 1 out of 2


Inequality Notation:
Number Line:

Answer 1

To solve the inequalities \(-2x + 1 < -9\) and \(-2x + 1 \geq -13\), we'll solve each inequality separately.

Solve the first inequality:

\[ -2x + 1 < -9 \] Subtract 1 from both sides: \[ -2x < -10 \] Now divide by -2, and remember to flip the inequality sign: \[ x > 5 \]

Solve the second inequality:

\[ -2x + 1 \geq -13 \] Subtract 1 from both sides: \[ -2x \geq -14 \] Now divide by -2, and flip the inequality sign: \[ x \leq 7 \]

Combine the solutions:

From the first inequality, we have \(x > 5\). From the second inequality, we have \(x \leq 7\).

Thus, the solution set is: \[ 5 < x \leq 7 \]

Inequality Notation:

The solution in interval notation is: \[ (5, 7] \]

Number Line Graph:

To represent this on a number line:

• Draw a number line and mark points at 5 and 7.
• Use an open circle at 5 (indicating that 5 is not included) and a closed circle at 7 (indicating that 7 is included).
• Shade the region between 5 and 7.

Would you like to visualize a sketch of the number line, or is the description sufficient?

Answer 2

Solve for xx and graph the solution on the number line below.
3, is less than or equal to, x, minus, 5, and, 6, is greater than or equal to, x, minus, 5
3≤x−5and
6≥x−5
Answer
Attempt 1 out of 2


Inequality Notation:
Number Line:

Answer 3

Let's solve the inequalities \(3 \leq x - 5\) and \(6 \geq x - 5\) step by step.

Solve the first inequality:

\[ 3 \leq x - 5 \] Add 5 to both sides: \[ 3 + 5 \leq x \] \[ 8 \leq x \] This can be written as: \[ x \geq 8 \]

Solve the second inequality:

\[ 6 \geq x - 5 \] Add 5 to both sides: \[ 6 + 5 \geq x \] \[ 11 \geq x \] This can be written as: \[ x \leq 11 \]

Combine the solutions:

From the first inequality, we have \(x \geq 8\). From the second inequality, we have \(x \leq 11\).

Thus, the overall solution set is: \[ 8 \leq x \leq 11 \]

Inequality Notation:

The solution in interval notation is: \[ [8, 11] \]

Number Line Graph:

To represent this on a number line:

1. Draw a number line and mark points at 8 and 11.
2. Use a closed circle at 8 (indicating that 8 is included) and a closed circle at 11 (indicating that 11 is included).
3. Shade the region between 8 and 11, including the endpoints.

Summary:

• Inequality Notation: \( [8, 11] \)
• Number Line: Include closed circles at 8 and 11, with shading between them.

Would you like any further assistance or visual representation?

Answer 4

For which values of x is the expression undefined?
start fraction, 4, x, minus, 28, divided by, x, plus, 5, end fraction
x+5
4x−28
​

Answer
Attempt 1 out of 2

Additional Solution
No Solution
x, equalsx=

Answer 5

To determine where the expression \(\frac{4x - 28}{x + 5}\) is undefined, we need to identify the values of \(x\) that make the denominator zero since a rational expression is undefined when its denominator equals zero.

Step 1: Set the denominator equal to zero

\[ x + 5 = 0 \]

Step 2: Solve for \(x\)

Subtract 5 from both sides: \[ x = -5 \]

Conclusion

The expression \(\frac{4x - 28}{x + 5}\) is undefined at \(x = -5\).

Final Answer

Values of \(x\) for which the expression is undefined: \[ x = -5 \]

If there is any more help or explanation you need, feel free to ask!

Answer 6

For which values of x is the expression undefined?
start fraction, x, squared, minus, 81, divided by, x, minus, 4, end fraction
x−4
x
2
−81

Answer 7

To determine where the expression \(\frac{x^2 - 81}{x - 4}\) is undefined, we need to find the values of \(x\) that make the denominator zero. A rational expression is undefined when its denominator is equal to zero.

Step 1: Set the denominator equal to zero

\[ x - 4 = 0 \]

Step 2: Solve for \(x\)

Add 4 to both sides: \[ x = 4 \]

Conclusion

The expression \(\frac{x^2 - 81}{x - 4}\) is undefined at \(x = 4\).

Final Answer

Values of \(x\) for which the expression is undefined: \[ x = 4 \]

If you have any further questions or need additional assistance, feel free to ask!

Answer 8

Solve for x and simplify the answer fully.
start fraction, 4, x, minus, 3, divided by, 2, x, minus, 7, end fraction, equals, start fraction, 5, divided by, 3, end fraction
2x−7
4x−3
​
=
3
5
​

Answer 9

To solve the equation

\[ \frac{4x - 3}{2x - 7} = \frac{5}{3} \]

we will cross-multiply to eliminate the fractions. The cross-multiplication gives us:

\[ 3(4x - 3) = 5(2x - 7) \]

Step 1: Expand both sides

Expanding the left-hand side: \[ 3(4x) - 3(3) = 12x - 9 \]

Expanding the right-hand side: \[ 5(2x) - 5(7) = 10x - 35 \]

Now our equation looks like: \[ 12x - 9 = 10x - 35 \]

Step 2: Rearrange the equation

Subtract \(10x\) from both sides: \[ 12x - 10x - 9 = -35 \] \[ 2x - 9 = -35 \]

Next, add 9 to both sides: \[ 2x = -35 + 9 \] \[ 2x = -26 \]

Step 3: Solve for \(x\)

Now, divide both sides by 2: \[ x = \frac{-26}{2} = -13 \]

Final Answer:

The solution is: \[ \boxed{-13} \]

The simplified answer for \(x\) is \(-13\). If you have any further questions or need more clarification, feel free to ask!

Answer 10

Solve for x and simplify the answer fully.
start fraction, 2, x, minus, 6, divided by, 7, end fraction, equals, start fraction, x, plus, 2, divided by, 6, end fraction
7
2x−6
​
=
6
x+2
​

Answer 11

To solve the equation

\[ \frac{2x - 6}{7} = \frac{x + 2}{6} \]

we will cross-multiply to eliminate the fractions. Cross-multiplication gives us:

\[ 6(2x - 6) = 7(x + 2) \]

Step 1: Expand both sides

Expanding the left-hand side: \[ 6 \cdot 2x - 6 \cdot 6 = 12x - 36 \]

Expanding the right-hand side: \[ 7 \cdot x + 7 \cdot 2 = 7x + 14 \]

Now the equation looks like: \[ 12x - 36 = 7x + 14 \]

Step 2: Rearrange the equation

First, subtract \(7x\) from both sides: \[ 12x - 7x - 36 = 14 \] \[ 5x - 36 = 14 \]

Next, add 36 to both sides: \[ 5x = 14 + 36 \] \[ 5x = 50 \]

Step 3: Solve for \(x\)

Now, divide both sides by 5: \[ x = \frac{50}{5} = 10 \]

Final Answer:

The solution is: \[ \boxed{10} \]

Thus, the simplified answer for \(x\) is \(10\). If you have any further questions or need additional assistance, feel free to ask!