Question
Steps for Solving Inequalities
Answers
GPT-4o mini
Solving inequalities involves a specific set of steps that are somewhat similar to solving equations, but with some important distinctions due to how inequalities work. Here's a general outline of the steps for solving inequalities:
### Steps for Solving Inequalities
1. **Understand the Inequality Symbols**:
- \( < \) means "less than"
- \( > \) means "greater than"
- \( \leq \) means "less than or equal to"
- \( \geq \) means "greater than or equal to"
- \( \neq \) means "not equal to"
2. **Isolate the Variable**:
- Start with the given inequality.
- Use algebraic operations to isolate the variable on one side of the inequality. You can add, subtract, multiply, or divide both sides, but be careful with multiplication and division.
3. **Reverse the Inequality Sign**:
- If you multiply or divide both sides of the inequality by a **negative number**, you must reverse the inequality sign.
4. **Simplify**:
- Simplify both sides of the inequality if possible. This helps to make the solution clearer.
5. **Express the Solution**:
- Write the solution in the form of an inequality or interval notation. For example, if \( x > 3 \), you can express it as \( (3, \infty) \).
6. **Graph the Solution (if required)**:
- If the context requires or if it helps you visualize, graph the solution on a number line:
- Use an open dot for “less than” or “greater than” (e.g., \( < \) or \( > \)).
- Use a closed dot for “less than or equal to” or “greater than or equal to” (e.g., \( \leq \) or \( \geq \)).
7. **Check Your Solution**:
- Select a test point from your solution set and substitute it back into the original inequality to verify that it satisfies the inequality. Also, test a point outside the solution set to confirm it does not satisfy the inequality.
### Example
Let’s solve the inequality \( -2x + 4 > 10 \).
1. **Isolate the variable**:
\[
-2x + 4 > 10
\]
Subtract \( 4 \) from both sides:
\[
-2x > 6
\]
2. **Divide by -2** (note that we need to reverse the inequality sign):
\[
x < -3
\]
3. **Express the solution**:
\[
x < -3
\]
In interval notation, this is \( (-\infty, -3) \).
4. **Check the solution**:
Test a point like \( -4 \):
\[
-2(-4) + 4 > 10 \rightarrow 8 + 4 > 10 \rightarrow 12 > 10 \quad (True)
\]
Test a point like \( -2 \):
\[
-2(-2) + 4 > 10 \rightarrow 4 + 4 > 10 \rightarrow 8 > 10 \quad (False)
\]
This concludes the process of solving the inequality. Always remember to be cautious about changing the direction of the inequality when multiplying or dividing by negative numbers!
### Steps for Solving Inequalities
1. **Understand the Inequality Symbols**:
- \( < \) means "less than"
- \( > \) means "greater than"
- \( \leq \) means "less than or equal to"
- \( \geq \) means "greater than or equal to"
- \( \neq \) means "not equal to"
2. **Isolate the Variable**:
- Start with the given inequality.
- Use algebraic operations to isolate the variable on one side of the inequality. You can add, subtract, multiply, or divide both sides, but be careful with multiplication and division.
3. **Reverse the Inequality Sign**:
- If you multiply or divide both sides of the inequality by a **negative number**, you must reverse the inequality sign.
4. **Simplify**:
- Simplify both sides of the inequality if possible. This helps to make the solution clearer.
5. **Express the Solution**:
- Write the solution in the form of an inequality or interval notation. For example, if \( x > 3 \), you can express it as \( (3, \infty) \).
6. **Graph the Solution (if required)**:
- If the context requires or if it helps you visualize, graph the solution on a number line:
- Use an open dot for “less than” or “greater than” (e.g., \( < \) or \( > \)).
- Use a closed dot for “less than or equal to” or “greater than or equal to” (e.g., \( \leq \) or \( \geq \)).
7. **Check Your Solution**:
- Select a test point from your solution set and substitute it back into the original inequality to verify that it satisfies the inequality. Also, test a point outside the solution set to confirm it does not satisfy the inequality.
### Example
Let’s solve the inequality \( -2x + 4 > 10 \).
1. **Isolate the variable**:
\[
-2x + 4 > 10
\]
Subtract \( 4 \) from both sides:
\[
-2x > 6
\]
2. **Divide by -2** (note that we need to reverse the inequality sign):
\[
x < -3
\]
3. **Express the solution**:
\[
x < -3
\]
In interval notation, this is \( (-\infty, -3) \).
4. **Check the solution**:
Test a point like \( -4 \):
\[
-2(-4) + 4 > 10 \rightarrow 8 + 4 > 10 \rightarrow 12 > 10 \quad (True)
\]
Test a point like \( -2 \):
\[
-2(-2) + 4 > 10 \rightarrow 4 + 4 > 10 \rightarrow 8 > 10 \quad (False)
\]
This concludes the process of solving the inequality. Always remember to be cautious about changing the direction of the inequality when multiplying or dividing by negative numbers!
how to isolate the variable
GPT-4o mini
Isolating the variable is a crucial step in solving equations and inequalities. It involves manipulating the expression in order to get the variable by itself on one side of the equation or inequality. Here’s a step-by-step guide to isolating a variable:
### Steps to Isolate the Variable
1. **Identify the Variable**: Determine which variable you want to isolate. This is typically denoted by a letter (e.g., \( x \), \( y \), etc.).
2. **Simplify the Expression**:
- Combine like terms if there are any on the same side of the equation or inequality.
- Eliminate any parentheses by distributing if necessary.
3. **Move Constant Terms**:
- If there are any constants (numbers) being added or subtracted from the variable, move them to the other side of the equation or inequality.
- You do this by performing the opposite operation. For example:
- If the equation is \( x + 5 = 10 \), subtract \( 5 \) from both sides:
\[
x = 10 - 5
\]
- If the inequality is \( x + 3 < 7 \), subtract \( 3 \) from both sides:
\[
x < 7 - 3
\]
4. **Move Coefficient Terms**:
- If the variable has a coefficient (a number multiplied by the variable), divide or multiply both sides of the equation or inequality by that coefficient, remembering to flip the inequality sign if it’s a negative number.
- For example:
- If \( 2x = 8 \), divide both sides by \( 2 \):
\[
x = \frac{8}{2} \Rightarrow x = 4
\]
- For the inequality \( -3x > 9 \), divide both sides by \( -3 \) (flip the inequality sign):
\[
x < \frac{9}{-3} \Rightarrow x < -3
\]
5. **Combine Operations**:
- In cases where multiple operations are needed to isolate the variable, perform them in the correct order:
- First, move constants (add or subtract).
- Next, manage coefficients (multiply or divide).
6. **Check for Extraneous Solutions**: After isolating the variable and finding a solution, always check if the value obtained satisfies the original equation or inequality, especially if you have performed operations that could change the solution set (like multiplying or dividing by variables).
### Example: Isolating a Variable
Let’s work through an example:
#### Solve the equation:
\( 4x - 8 = 12 \)
1. **Move Constant Terms**:
- Add \( 8 \) to both sides:
\[
4x - 8 + 8 = 12 + 8
\]
This simplifies to:
\[
4x = 20
\]
2. **Move Coefficient Terms**:
- Divide both sides by \( 4 \):
\[
x = \frac{20}{4}
\]
Which simplifies to:
\[
x = 5
\]
#### Solve the inequality:
\( 3x + 4 \leq 13 \)
1. **Move Constant Terms**:
- Subtract \( 4 \) from both sides:
\[
3x + 4 - 4 \leq 13 - 4
\]
Simplifying gives us:
\[
3x \leq 9
\]
2. **Move Coefficient Terms**:
- Divide both sides by \( 3 \):
\[
x \leq 3
\]
### Conclusion
By following these steps, you can isolate the variable effectively in equations and inequalities. Just remember that the operations you perform must be consistent on both sides of the equation or inequality to maintain equality or the correct inequality relationship.
### Steps to Isolate the Variable
1. **Identify the Variable**: Determine which variable you want to isolate. This is typically denoted by a letter (e.g., \( x \), \( y \), etc.).
2. **Simplify the Expression**:
- Combine like terms if there are any on the same side of the equation or inequality.
- Eliminate any parentheses by distributing if necessary.
3. **Move Constant Terms**:
- If there are any constants (numbers) being added or subtracted from the variable, move them to the other side of the equation or inequality.
- You do this by performing the opposite operation. For example:
- If the equation is \( x + 5 = 10 \), subtract \( 5 \) from both sides:
\[
x = 10 - 5
\]
- If the inequality is \( x + 3 < 7 \), subtract \( 3 \) from both sides:
\[
x < 7 - 3
\]
4. **Move Coefficient Terms**:
- If the variable has a coefficient (a number multiplied by the variable), divide or multiply both sides of the equation or inequality by that coefficient, remembering to flip the inequality sign if it’s a negative number.
- For example:
- If \( 2x = 8 \), divide both sides by \( 2 \):
\[
x = \frac{8}{2} \Rightarrow x = 4
\]
- For the inequality \( -3x > 9 \), divide both sides by \( -3 \) (flip the inequality sign):
\[
x < \frac{9}{-3} \Rightarrow x < -3
\]
5. **Combine Operations**:
- In cases where multiple operations are needed to isolate the variable, perform them in the correct order:
- First, move constants (add or subtract).
- Next, manage coefficients (multiply or divide).
6. **Check for Extraneous Solutions**: After isolating the variable and finding a solution, always check if the value obtained satisfies the original equation or inequality, especially if you have performed operations that could change the solution set (like multiplying or dividing by variables).
### Example: Isolating a Variable
Let’s work through an example:
#### Solve the equation:
\( 4x - 8 = 12 \)
1. **Move Constant Terms**:
- Add \( 8 \) to both sides:
\[
4x - 8 + 8 = 12 + 8
\]
This simplifies to:
\[
4x = 20
\]
2. **Move Coefficient Terms**:
- Divide both sides by \( 4 \):
\[
x = \frac{20}{4}
\]
Which simplifies to:
\[
x = 5
\]
#### Solve the inequality:
\( 3x + 4 \leq 13 \)
1. **Move Constant Terms**:
- Subtract \( 4 \) from both sides:
\[
3x + 4 - 4 \leq 13 - 4
\]
Simplifying gives us:
\[
3x \leq 9
\]
2. **Move Coefficient Terms**:
- Divide both sides by \( 3 \):
\[
x \leq 3
\]
### Conclusion
By following these steps, you can isolate the variable effectively in equations and inequalities. Just remember that the operations you perform must be consistent on both sides of the equation or inequality to maintain equality or the correct inequality relationship.