Algebra 2 Practice - Solve and Graph an Inequality (Example 1)

To solve and graph an inequality, follow these steps:

1. **Isolate the Variable**: If there are any other terms present, isolate the variable on one side of the inequality. For example, if you have \(2x + 3\) is less than \(9\), isolate \(x\) by subtracting 3 from both sides to get \(2x\) is less than \(6\).

2. **Solve for the Variable**: Once the variable is isolated, solve the inequality to find the range of values for the variable that satisfy the inequality. In the example \(2x\) is less than \(6\), divide both sides by 2 to get \(x\) is less than \(3\).

3. **Graph the Solution on a Number Line**: Draw a horizontal line and mark it with numbers that cover the range of values relevant to the inequality. Place the solution(s) on the number line.

4. **Check for Inequality Direction**: If the inequality is "is less than" or "is greater than", use an open circle to represent the endpoint if the inequality is strict. If it's "is less than or equal to" or "is greater than or equal to", use a closed circle to represent the endpoint.

5. **Shade the Appropriate Region**: Shade the region on the number line that corresponds to the solution set. If the inequality is "is less than" or "is greater than", shade the line in the direction that satisfies the inequality. If it's "is less than or equal to" or "is greater than or equal to", shade the line including the marked point.

6. **Label the Shaded Region**: If necessary, label the shaded region to indicate which side of the inequality they represent.

7. **Check for Infinity**: If the solution includes all real numbers greater than a certain value (or less than a certain value), use an arrow to show that the solution extends indefinitely in that direction.

8. **Represent Multiple Inequalities**: If there are multiple inequalities, graph each one separately on the same number line and consider the intersection of their solution sets.

That's it! You've solved and graphed the inequality. Remember, the solution set consists of all the values that make the inequality true.

Remember, when solving absolute value inequalities, you may end up with multiple intervals of solutions. Each interval needs to be represented separately on the number line. Also, be mindful of the direction of the inequality to determine whether to use open or closed circles and which side to shade.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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