solve 10x+4<8x-8

Question

then place it on a number line

Answer 1

To solve the inequality \(10x + 4 < 8x - 8\), we will isolate \(x\).

1. Subtract \(8x\) from both sides:

\[ 10x - 8x + 4 < -8 \]

This simplifies to:

\[ 2x + 4 < -8 \]

2. Next, subtract 4 from both sides:

\[ 2x < -8 - 4 \]

This simplifies to:

\[ 2x < -12 \]

3. Now, divide both sides by 2:

\[ x < -6 \]

Conclusion

The solution to the inequality is:

\[ x < -6 \]

Number Line Representation

To represent this on a number line:

1. Draw a horizontal line.
2. Mark a point at \(-6\).
3. Draw an open circle at \(-6\) (indicating that \(-6\) is not included).
4. Shade the line to the left of \(-6\) to indicate all the values less than \(-6\).

Here's a simple representation (text-based):

<---|---|---|---|---|---|---|---|---|---|---|---|--->
        -7  -6  -5  -4  -3  -2  -1   0   1   2
          (--->

The open parenthesis at \(-6\) indicates that this value is not included in the solution.

Answer 2

what is the solution set of x+2≥6 and 3x≥6

Answer 3

To find the solution set for the inequalities \(x + 2 \geq 6\) and \(3x \geq 6\), we will solve each inequality separately.

First Inequality: \(x + 2 \geq 6\)

1. Subtract 2 from both sides: \[ x \geq 6 - 2 \]
2. This simplifies to: \[ x \geq 4 \]

Second Inequality: \(3x \geq 6\)

1. Divide both sides by 3: \[ x \geq \frac{6}{3} \]
2. This simplifies to: \[ x \geq 2 \]

Combined Solution

Now, we need to find the intersection of the two solution sets:

1. From the first inequality, we have \(x \geq 4\).
2. From the second inequality, we have \(x \geq 2\).

Since \(x \geq 4\) is stronger than \(x \geq 2\), the solution set for the combined inequalities is:

\[ x \geq 4 \]

Interval Notation

The solution set in interval notation is:

\[ [4, \infty) \]

Conclusion

Thus, the solution set is \(x \geq 4\), which means all values of \(x\) that are 4 or greater.

Answer 4

what is the solution set of 2x-3=5 and 5x+3>-12

Answer 5

To find the solution set for the equations and inequalities \(2x - 3 = 5\) and \(5x + 3 > -12\), we will solve each one separately.

First Equation: \(2x - 3 = 5\)

1. Add 3 to both sides: \[ 2x = 5 + 3 \] \[ 2x = 8 \]

2. Divide both sides by 2: \[ x = \frac{8}{2} \] \[ x = 4 \]

Second Inequality: \(5x + 3 > -12\)

1. Subtract 3 from both sides: \[ 5x > -12 - 3 \] \[ 5x > -15 \]

2. Divide both sides by 5: \[ x > -3 \]

Combined Solution

Now we summarize the solutions:

1. From the first equation, we found \(x = 4\).
2. From the second inequality, we found \(x > -3\).

Conclusion

• The solution to the equation \(2x - 3 = 5\) is \(x = 4\).
• The solution to the inequality \(5x + 3 > -12\) is \(x > -3\).

Since \(4 > -3\), the solution set of the inequalities includes \(x = 4\).

Therefore:

• The solution to the system is simply \( { 4 } \) for the equation and \( (-3, \infty) \) for the inequality.
• The value \(x = 4\) is also a valid solution to the greater inequality \(x > -3\).

Final Solution Set

The combined solution set, including both the equation and the inequality, is:

\[ { 4 } \text{ and } (-3, \infty) \]

In summary, \(x = 4\) satisfies both conditions.

Answer 6

what is the solution set of 2x>-8 and -5x+7=12

Answer 7

To determine the solution set for the inequalities and equations \(2x > -8\) and \(-5x + 7 = 12\), we will solve each one separately.

First Inequality: \(2x > -8\)

1. Divide both sides by 2: \[ x > \frac{-8}{2} \]
2. This simplifies to: \[ x > -4 \]

Second Equation: \(-5x + 7 = 12\)

1. Subtract 7 from both sides: \[ -5x = 12 - 7 \] \[ -5x = 5 \]

2. Divide both sides by -5 (remember to reverse the inequality when dividing by a negative number): \[ x = \frac{5}{-5} \] \[ x = -1 \]

Combined Solution

Now we summarize the solutions:

1. From the first inequality \(2x > -8\), we found \(x > -4\).
2. From the equation \(-5x + 7 = 12\), we found \(x = -1\).

Conclusion

• The solution to the inequality is \(x > -4\).
• The solution to the equation is \(x = -1\).

Intersection of Solutions

Since \(-1\) is greater than \(-4\), it satisfies the inequality \(x > -4\).

Final Solution Set

Thus, the solution set is:

\[ (-4, \infty) \text{ (from the inequality) and } {-1} \text{ (from the equation)} \]

Since \(x = -1\) falls within the interval where \(x > -4\), it is included in the solution set.

So the final solution set can be expressed as:

\[ (-4, \infty) \]

Or, if you wish to highlight both parts, it can be represented as:

\[ {-1} \cup (-4, \infty) \]

This indicates that \(x = -1\) is part of the solutions for the equation and that the range continues to include values greater than \(-4\).

Answer 8

what is the solution set of -3x+7=10 or 3x-9=0

Answer 9

To find the solution set for the equations \(-3x + 7 = 10\) and \(3x - 9 = 0\), we will solve each one separately.

First Equation: \(-3x + 7 = 10\)

1. Subtract 7 from both sides: \[ -3x = 10 - 7 \] \[ -3x = 3 \]

2. Divide both sides by -3: \[ x = \frac{3}{-3} \] \[ x = -1 \]

Second Equation: \(3x - 9 = 0\)

1. Add 9 to both sides: \[ 3x = 9 \]

2. Divide both sides by 3: \[ x = \frac{9}{3} \] \[ x = 3 \]

Combined Solution

Now we summarize the solutions:

1. From the first equation \(-3x + 7 = 10\), we found \(x = -1\).
2. From the second equation \(3x - 9 = 0\), we found \(x = 3\).

Conclusion

Since the problem uses 'or', the complete solution set includes both solutions.

Final Solution Set

\[ {-1, 3} \]

This means that the solution set consists of two separate values: \(x = -1\) and \(x = 3\).