Describe in your own words how to solve a linear equation using the equality properties. Demonstrate the process with an example. Next, replace the equal sign in your example with an inequality by using the less than or greater than sign. Then solve the inequality.
The process in solving an inequality is the same as for an equation, except for multiplying/dividing by negative quantities, when the carat (< or >) reverses.
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To solve a linear equation using the equality properties, follow these steps:
1. Start by gathering all the terms with variables on one side of the equation and all the constants on the other side.
2. Simplify both sides of the equation as much as possible by combining like terms.
3. Use the appropriate equality properties to isolate the variable on one side of the equation.
4. Divide both sides of the equation by the coefficient of the variable to solve for the variable.
Let's demonstrate this process with the example equation: 3x + 2 = 8.
1. Move the constant term (2) to the other side of the equation by subtracting it from both sides: 3x = 8 - 2, which simplifies to 3x = 6.
2. There are no like terms on either side, so the equation remains as 3x = 6.
3. To isolate the variable x, we can use the property of multiplication and divide both sides of the equation by the coefficient (3): (3x) / 3 = 6 / 3, which simplifies to x = 2.
4. Therefore, the solution to the equation 3x + 2 = 8 is x = 2.
Now, let's replace the equal sign in this example with an inequality sign, such as the greater than sign (>).
The inequality would then be: 3x + 2 > 8.
To solve this inequality, we follow similar steps:
1. Move the constant term (2) to the other side by subtracting it from both sides: 3x > 8 - 2, which simplifies to 3x > 6.
2. There are no like terms on either side, so the inequality remains as 3x > 6.
3. To isolate the variable x, we need to divide both sides of the inequality by the coefficient (3). However, when dividing by a negative number (which is the case with 3), we need to flip the inequality sign. So we have: (3x) / 3 < 6 / 3, which simplifies to x < 2.
4. Therefore, the solution to the inequality 3x + 2 > 8 is x < 2.
Note that when solving inequalities, if we were to multiply or divide both sides by a negative number, we would need to flip the inequality sign.