How Do You Solve Multi Step Inequalities

Solving multi-step inequalities requires a combination of algebraic techniques used for solving equations, with an added understanding of how inequalities behave. The goal is to isolate the variable on one side of the inequality to determine the range of values that satisfy the statement. This comprehensive guide will walk you through the steps, providing clear examples and explanations to master this essential skill.

Understanding Inequalities

Before diving into multi-step inequalities, it's crucial to understand the basics of inequalities themselves. An inequality compares two values, showing that they are not necessarily equal.

• Symbols Used:

  • < : Less than
  • > : Greater than
  • ≤ : Less than or equal to
  • ≥ : Greater than or equal to
  • ≠ : Not equal to

• Basic Properties:

  • Adding or subtracting the same number from both sides does not change the inequality.
  • Multiplying or dividing both sides by a positive number does not change the inequality.
  • Multiplying or dividing both sides by a negative number reverses the direction of the inequality. This is a critical rule to remember!

Steps to Solve Multi-Step Inequalities

Solving multi-step inequalities involves a systematic approach. Here’s a breakdown of the steps involved:

1. Simplify Both Sides:
   • Combine like terms on each side of the inequality.
   • Distribute any multiplication over parentheses.
2. Isolate the Variable Term:
   • Use addition or subtraction to move terms without the variable to the other side of the inequality.
3. Isolate the Variable:
   • Use multiplication or division to isolate the variable.
   • Remember to flip the inequality sign if you multiply or divide by a negative number.
4. Check Your Solution:
   • Substitute a value from your solution set back into the original inequality to ensure it holds true.
5. Graph the Solution (Optional but Recommended):
   • Represent the solution set on a number line.

Detailed Explanation with Examples

Let's illustrate these steps with several examples.

Example 1: A Simple Multi-Step Inequality

Solve: 3x + 5 < 14

1. Simplify: Both sides are already simplified.
2. Isolate the Variable Term: Subtract 5 from both sides: 3x + 5 - 5 < 14 - 5 3x < 9
3. Isolate the Variable: Divide both sides by 3: 3x / 3 < 9 / 3 x < 3
4. Check: Let's pick x = 2 (which is less than 3) and substitute it into the original inequality: 3(2) + 5 < 14 6 + 5 < 14 11 < 14 (True)
5. Graph: Draw a number line. Place an open circle at 3 (because x is strictly less than 3) and shade the line to the left, indicating all values less than 3.

Example 2: Inequality with Distribution

Solve: 2(x - 1) ≥ 5x + 4

1. Simplify: Distribute the 2 on the left side: 2x - 2 ≥ 5x + 4
2. Isolate the Variable Term: Subtract 2x from both sides: 2x - 2 - 2x ≥ 5x + 4 - 2x -2 ≥ 3x + 4 Subtract 4 from both sides: -2 - 4 ≥ 3x + 4 - 4 -6 ≥ 3x
3. Isolate the Variable: Divide both sides by 3: -6 / 3 ≥ 3x / 3 -2 ≥ x (This can also be written as x ≤ -2)
4. Check: Let's pick x = -3 (which is less than or equal to -2) and substitute it into the original inequality: 2(-3 - 1) ≥ 5(-3) + 4 2(-4) ≥ -15 + 4 -8 ≥ -11 (True)
5. Graph: Draw a number line. Place a closed circle at -2 (because x is less than or equal to -2) and shade the line to the left, indicating all values less than or equal to -2.

Example 3: Inequality with a Negative Coefficient

Solve: -4x + 7 > 15

1. Simplify: Both sides are already simplified.
2. Isolate the Variable Term: Subtract 7 from both sides: -4x + 7 - 7 > 15 - 7 -4x > 8
3. Isolate the Variable: Divide both sides by -4. Remember to flip the inequality sign! -4x / -4 < 8 / -4 x < -2
4. Check: Let's pick x = -3 (which is less than -2) and substitute it into the original inequality: -4(-3) + 7 > 15 12 + 7 > 15 19 > 15 (True)
5. Graph: Draw a number line. Place an open circle at -2 (because x is strictly less than -2) and shade the line to the left, indicating all values less than -2.

Example 4: Inequality with Fractions

Solve: (2/3)x - 1 ≤ 5

1. Simplify: Both sides are already simplified.
2. Isolate the Variable Term: Add 1 to both sides: (2/3)x - 1 + 1 ≤ 5 + 1 (2/3)x ≤ 6
3. Isolate the Variable: Multiply both sides by the reciprocal of 2/3, which is 3/2: (3/2) * (2/3)x ≤ 6 * (3/2) x ≤ 9
4. Check: Let's pick x = 0 (which is less than or equal to 9) and substitute it into the original inequality: (2/3)(0) - 1 ≤ 5 0 - 1 ≤ 5 -1 ≤ 5 (True)
5. Graph: Draw a number line. Place a closed circle at 9 (because x is less than or equal to 9) and shade the line to the left, indicating all values less than or equal to 9.

Example 5: Inequality with Variables on Both Sides and Distribution

Solve: 3(2x + 1) < 4x - 5

1. Simplify: Distribute the 3 on the left side: 6x + 3 < 4x - 5
2. Isolate the Variable Term: Subtract 4x from both sides: 6x + 3 - 4x < 4x - 5 - 4x 2x + 3 < -5 Subtract 3 from both sides: 2x + 3 - 3 < -5 - 3 2x < -8
3. Isolate the Variable: Divide both sides by 2: 2x / 2 < -8 / 2 x < -4
4. Check: Let's pick x = -5 (which is less than -4) and substitute it into the original inequality: 3(2(-5) + 1) < 4(-5) - 5 3(-10 + 1) < -20 - 5 3(-9) < -25 -27 < -25 (True)
5. Graph: Draw a number line. Place an open circle at -4 (because x is strictly less than -4) and shade the line to the left, indicating all values less than -4.

Example 6: A More Complex Inequality

Solve: 5x - 3(x + 2) > 4(x - 1) + 7

1. Simplify: Distribute on both sides: 5x - 3x - 6 > 4x - 4 + 7 2x - 6 > 4x + 3
2. Isolate the Variable Term: Subtract 2x from both sides: 2x - 6 - 2x > 4x + 3 - 2x -6 > 2x + 3 Subtract 3 from both sides: -6 - 3 > 2x + 3 - 3 -9 > 2x
3. Isolate the Variable: Divide both sides by 2: -9 / 2 > 2x / 2 -4.5 > x (This can also be written as x < -4.5)
4. Check: Let's pick x = -5 (which is less than -4.5) and substitute it into the original inequality: 5(-5) - 3(-5 + 2) > 4(-5 - 1) + 7 -25 - 3(-3) > 4(-6) + 7 -25 + 9 > -24 + 7 -16 > -17 (True)
5. Graph: Draw a number line. Place an open circle at -4.5 (because x is strictly less than -4.5) and shade the line to the left, indicating all values less than -4.5.

Special Cases

There are two special cases you might encounter when solving inequalities:

1. No Solution: The inequality simplifies to a false statement, regardless of the value of the variable. For example:

   Solve: 2x + 3 > 2x + 5 Subtract 2x from both sides: 3 > 5 (False) In this case, there is no solution.

2. All Real Numbers: The inequality simplifies to a true statement, regardless of the value of the variable. For example:

   Solve: 3x - 1 < 3x + 4 Subtract 3x from both sides: -1 < 4 (True) In this case, the solution is all real numbers.

Tips for Success

• Pay Attention to the Inequality Sign: Always be mindful of the direction of the inequality sign and remember to flip it when multiplying or dividing by a negative number.
• Double-Check Your Work: Mistakes are easy to make, especially with multiple steps. Take the time to review your work carefully.
• Use the Check Step: The check step is crucial to ensure that your solution is correct. It helps you catch any errors you might have made.
• Practice Regularly: The more you practice, the more comfortable you will become with solving inequalities.

Real-World Applications

Inequalities are used in many real-world applications, such as:

• Budgeting: Determining how much you can spend on different items while staying within your budget.
• Manufacturing: Ensuring that products meet certain quality control standards.
• Science: Modeling physical phenomena and making predictions.
• Optimization: Finding the best possible solution to a problem, such as maximizing profit or minimizing costs.

Conclusion

Solving multi-step inequalities is a fundamental skill in algebra. By following the steps outlined in this guide and practicing regularly, you can master this skill and confidently tackle more complex problems. Remember to pay close attention to the inequality sign and always check your solution to ensure accuracy. With practice and perseverance, you can become proficient in solving multi-step inequalities and apply this knowledge to various real-world situations.