Multi-step inequalities with decimals

Key Notes:

1. Understanding Inequalities

• An inequality is a mathematical statement that shows the relationship between two expressions using symbols like <, >, ≤, and ≥.
• Examples:
  • 3.5 > 2.1 (3.5 is greater than 2.1)
  • x ≤ 4.6 (x is less than or equal to 4.6)

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2. Solving Multi-step Inequalities

• Multi-step inequalities involve more than one operation (addition, subtraction, multiplication, or division) to solve for the variable.
• The steps may include combining like terms, adding/subtracting numbers, or dividing/multiplying both sides of the inequality.

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3. Steps to Solve Multi-step Inequalities with Decimals

• Step 1: Simplify both sides of the inequality if needed (combine like terms or distribute).
• Step 2: Isolate the variable (get the variable on one side of the inequality).
• Step 3: If necessary, perform operations like adding or subtracting decimals on both sides.
• Step 4: If you multiply or divide by a negative number, flip the inequality sign.
• Step 5: Solve for the variable and express the solution.

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4. Example 1: Solve the inequality:

• 0.5x + 2.1 ≥ 4.6
  1. Subtract 2.1 from both sides:
     0.5x ≥ 2.5
  2. Divide both sides by 0.5:
     x ≥ 5

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5. Example 2: Solve the inequality:

• 1.2x – 3.4 < 5.6
  1. Add 3.4 to both sides:
     1.2x < 9
  2. Divide both sides by 1.2:
     x < 7.5

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6. Tips for Working with Decimals in Inequalities

• Convert decimals to fractions (if easier for you to work with) and then solve.
• Keep track of the decimal places during addition, subtraction, multiplication, and division.
• Use a calculator when necessary to ensure accuracy, especially for non-integer results.

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7. Checking Your Solution

• After solving, check your solution by substituting the value of the variable back into the original inequality to see if it satisfies the inequality.

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8. Graphing Solutions

• After solving an inequality, you can represent the solution on a number line.
• Use an open circle for < or > and a closed circle for ≤ or ≥.

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9. Practice Problems

• (a) Solve: 2.5x + 3.2 ≤ 8.7
• (b) Solve: 0.6x – 1.4 > 3.5
• (c) Solve: 4.8x + 2.1 ≥ 10.5

Learn with an example

Let’s practice!