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.
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To solve a linear equation using the equality properties, you need to follow these steps:
1. Start with the given equation and identify the variable you want to isolate. The goal is to have the variable on one side of the equation and constants on the other side.
2. Use the equality properties to perform operations that will simplify the equation and eventually isolate the variable. These properties include adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
3. Continue simplifying the equation by combining like terms and applying the distributive property, if needed.
4. Finally, solve for the variable by performing the inverse operation necessary to isolate the variable. This will give you the solution to the equation.
Let's demonstrate this with an example:
Example: Solve the equation 3x - 5 = 7.
1. Identify the variable, which is x, and isolate it on one side of the equation.
3x - 5 = 7
2. Use the equality properties to simplify the equation. Since we want to isolate x, we will start by adding 5 on both sides to get rid of the -5 term.
3x - 5 + 5 = 7 + 5
3x = 12
3. Simplify the equation further by combining like terms.
3x = 12
4. Finally, to isolate x, divide both sides by the coefficient of x, which is 3.
(3x)/3 = 12/3
x = 4
Therefore, the solution to the equation 3x - 5 = 7 is x = 4.
Now, let's replace the equal sign in our example equation with an inequality sign (less than or greater than) and solve the inequality.
Example: Solve the inequality 3x - 5 > 7.
1. Follow the same steps as in solving an equation, but note that when multiplying or dividing by a negative number, the inequality sign should be flipped.
3x - 5 > 7
2. Add 5 on both sides to get rid of the -5 term.
3x - 5 + 5 > 7 + 5
3x > 12
3. Simplify the inequality further.
3x > 12
4. To isolate x, divide both sides by the coefficient of x, which is 3. Remember to flip the inequality sign when dividing by a negative number (3 is positive).
(3x)/3 > 12/3
x > 4
Therefore, the solution to the inequality 3x - 5 > 7 is x > 4.