Solve Each Compound Inequality And Graph Its Solution

Solve Each Compound Inequality and Graph Its Solution: A Comprehensive Guide

Compound inequalities involve two or more inequalities joined by the words "and" or "or." Solving and graphing these inequalities requires a nuanced understanding of inequality rules and number line representation. This comprehensive guide will walk you through the process, equipping you with the skills to tackle any compound inequality problem.

Understanding Compound Inequalities

Compound inequalities combine two simple inequalities using "and" or "or."

• "And" Inequalities: A solution satisfies an "and" inequality only if it satisfies both individual inequalities simultaneously. The solution set is the intersection of the solution sets of the individual inequalities.

• "Or" Inequalities: A solution satisfies an "or" inequality if it satisfies at least one of the individual inequalities. The solution set is the union of the solution sets of the individual inequalities.

Solving "And" Compound Inequalities

Let's explore how to solve and graph "and" compound inequalities. The general form is: a < x < b or a ≤ x ≤ b

Example 1: Solve and graph the inequality: -2 < x + 3 ≤ 5

Solution:

1. Isolate the variable: To isolate 'x', subtract 3 from all parts of the inequality: -2 - 3 < x + 3 - 3 ≤ 5 - 3 -5 < x ≤ 2

2. Graph the solution: On a number line, represent the solution. Use an open circle for < or > and a closed circle for ≤ or ≥. In this case, we have an open circle at -5 and a closed circle at 2, with a line connecting them, indicating all values between -5 and 2 (inclusive of 2).

(Insert image here: A number line with an open circle at -5, a closed circle at 2, and a line connecting them.)

Example 2: Solve and graph: 2x - 1 ≥ 3 and x + 4 < 7

Solution:

1. Solve each inequality separately:

   • 2x - 1 ≥ 3 => 2x ≥ 4 => x ≥ 2
   • x + 4 < 7 => x < 3

2. Find the intersection: The solution must satisfy both inequalities. This means x must be greater than or equal to 2 and less than 3.

3. Graph the solution: The graph will show a closed circle at 2 and an open circle at 3, with a line connecting them. This represents the values between 2 (inclusive) and 3 (exclusive).

(Insert image here: A number line with a closed circle at 2, an open circle at 3, and a line connecting them.)

Solving "Or" Compound Inequalities

"Or" inequalities are solved similarly, but the final step involves finding the union of the solution sets.

Example 3: Solve and graph: x - 2 > 1 or x + 5 ≤ 0

Solution:

1. Solve each inequality separately:

   • x - 2 > 1 => x > 3
   • x + 5 ≤ 0 => x ≤ -5

2. Find the union: The solution includes values that satisfy either inequality. This is all values greater than 3 or less than or equal to -5.

3. Graph the solution: The graph will have an open circle at 3 with a line extending to the right and a closed circle at -5 with a line extending to the left.

(Insert image here: A number line with an open circle at 3, a line extending to the right, a closed circle at -5 and a line extending to the left.)

Example 4: A more complex "or" inequality: 3x + 2 < -1 or 2x - 5 ≥ 1

Solution:

1. Solve each inequality:

   • 3x + 2 < -1 => 3x < -3 => x < -1
   • 2x - 5 ≥ 1 => 2x ≥ 6 => x ≥ 3

2. Find the union: The solution set includes all values less than -1 or greater than or equal to 3.

3. Graph the solution: The graph will display an open circle at -1 with a line extending to the left and a closed circle at 3 with a line extending to the right.

(Insert image here: A number line with an open circle at -1, a line extending to the left, a closed circle at 3, and a line extending to the right.)

Special Cases and Considerations

No Solution: Some compound inequalities, particularly "and" inequalities, may have no solution. For example, x < 2 and x > 3 has no solution because no number can be simultaneously less than 2 and greater than 3. The graph would be empty.

All Real Numbers: Conversely, some "or" inequalities have a solution set encompassing all real numbers. For example, x < 5 or x > 2 includes all real numbers. The graph would be the entire number line.

Solving Compound Inequalities with Absolute Values

Absolute value inequalities introduce another layer of complexity. Recall that the absolute value of a number is its distance from zero.

Example 5: Solve and graph: |x - 2| < 3

Solution:

This inequality means the distance between x and 2 is less than 3. This translates to:

-3 < x - 2 < 3

Now solve as a regular "and" inequality:

-1 < x < 5

The graph will show an open circle at -1 and 5 with a line connecting them.

(Insert image here: A number line with open circles at -1 and 5, and a line connecting them.)

Example 6: Solve and graph: |2x + 1| ≥ 5

Solution:

This inequality represents cases where the distance between 2x + 1 and 0 is greater than or equal to 5. This translates to two separate inequalities:

2x + 1 ≥ 5 or 2x + 1 ≤ -5

Solve each separately:

2x ≥ 4 => x ≥ 2 2x ≤ -6 => x ≤ -3

The graph will show a closed circle at 2 with a line extending to the right, and a closed circle at -3 with a line extending to the left.

(Insert image here: A number line with a closed circle at 2, a line extending to the right, a closed circle at -3, and a line extending to the left.)

Real-World Applications of Compound Inequalities

Compound inequalities appear frequently in real-world scenarios, often involving constraints or limitations. Consider these examples:

• Temperature ranges: The temperature must stay between 65°F and 75°F. This is represented as 65 ≤ T ≤ 75.

• Speed limits: A car's speed must be at least 40 mph but no more than 65 mph. This can be written as 40 ≤ s ≤ 65.

Advanced Compound Inequalities

Solving more complex compound inequalities might involve multiple steps or require combining different techniques learned above. For instance, you might encounter inequalities containing fractions, exponents, or combinations of "and" and "or" conditions within the same problem. The key is to break down the problem into smaller, more manageable parts. Remember to always follow the order of operations (PEMDAS/BODMAS) and apply the properties of inequalities consistently.

Conclusion

Mastering compound inequalities requires a solid grasp of basic inequality principles, the ability to solve multiple inequalities concurrently, and a clear understanding of the concepts of intersection and union. By following the steps outlined in this guide and practicing regularly, you can build confidence and proficiency in solving and graphing any compound inequality. Remember to always carefully consider the meaning of "and" and "or" to correctly interpret and represent the solution sets on a number line. Through diligent practice, you will be well-equipped to tackle even the most challenging compound inequality problems.