solving multi step inequalities worksheet

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05-12-2024

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Meta Description: Conquer multi-step inequalities! This guide provides a step-by-step approach, examples, and practice problems to master solving inequalities, including those with variables on both sides and parentheses. Perfect for students and anyone looking to improve their algebra skills.

Understanding Inequalities

Before diving into multi-step inequalities, let's review the basics. Inequalities show the relationship between two expressions that are not equal. We use symbols like:

• <: less than
• >: greater than
• ≤: less than or equal to
• ≥: greater than or equal to

Solving an inequality means finding all the values of the variable that make the inequality true.

Solving Multi-Step Inequalities: A Step-by-Step Approach

Solving multi-step inequalities involves applying the same principles as solving multi-step equations, with one crucial difference: when you multiply or divide by a negative number, you must reverse the inequality symbol.

Here's a step-by-step approach:

1. Simplify each side: Combine like terms on each side of the inequality.

2. Isolate the variable term: Use addition or subtraction to move all terms with the variable to one side of the inequality and all constant terms to the other side.

3. Solve for the variable: Use multiplication or division to isolate the variable. Remember to reverse the inequality symbol if you multiply or divide by a negative number.

4. Graph the solution: Represent the solution set on a number line. Use an open circle (○) for < or > and a closed circle (●) for ≤ or ≥.

Example 1: Solving a Multi-Step Inequality

Let's solve the inequality: 3x + 5 > 11

1. Simplify: The left side is already simplified.

2. Isolate: Subtract 5 from both sides: 3x > 6

3. Solve: Divide both sides by 3: x > 2

4. Graph: The solution is all values greater than 2. The graph would show an open circle at 2 and an arrow pointing to the right.

Example 2: Inequality with Variables on Both Sides

Solve: 2x - 7 ≤ 5x + 8

1. Simplify: Both sides are simplified.

2. Isolate: Subtract 2x from both sides: -7 ≤ 3x + 8 Then subtract 8 from both sides: -15 ≤ 3x

3. Solve: Divide both sides by 3: -5 ≤ x This is the same as x ≥ -5

4. Graph: The graph would show a closed circle at -5 and an arrow pointing to the right.

Example 3: Inequality with Parentheses

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

1. Simplify: Distribute the 2 on the left side: 2x + 6 < 4x - 2

2. Isolate: Subtract 2x from both sides: 6 < 2x - 2 Add 2 to both sides: 8 < 2x

3. Solve: Divide both sides by 2: 4 < x This is the same as x > 4

4. Graph: The graph would show an open circle at 4 and an arrow pointing to the right.

Common Mistakes to Avoid

• Forgetting to reverse the inequality symbol: This is the most common mistake when multiplying or dividing by a negative number.

• Incorrectly combining like terms: Double-check your addition and subtraction.

• Errors in distributing: Be careful when distributing negative numbers.

Practice Problems

Here are some practice problems to solidify your understanding:

1. 5x - 10 > 25
2. -2x + 7 ≤ 13
3. 4(x - 2) ≥ 6x + 8
4. 3x + 5 < 2x - 1
5. -x + 6 > 2x - 3

Solutions to Practice Problems

1. x > 7
2. x ≥ -3
3. x ≤ -10
4. x < -6
5. x < 3

This comprehensive guide should give you a solid foundation in solving multi-step inequalities. Remember to practice regularly to build confidence and proficiency. Good luck!