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What is an example of an equivalent inequality?

Published in Inequality Examples 2 mins read

An example of an equivalent inequality is 2x - 3 > 7 and x > 5.

Understanding Equivalent Inequalities

Equivalent inequalities are inequalities that have the same solution set. This means that any value of the variable that satisfies one inequality will also satisfy the other. We obtain equivalent inequalities by applying the same operations to both sides of an inequality, similar to how we solve equations.

Example and Explanation

Let's consider the inequality from the provided reference:

  • Original Inequality: 2x - 3 > 7

This inequality states that "two times a number x, minus three, is greater than seven." To find the values of x that satisfy this condition, we need to isolate x. We can do this by performing the following steps:

  1. Add 3 to both sides: This cancels out the "-3" on the left side.

    2x - 3 + 3 > 7 + 3

    This simplifies to:

    2x > 10

  2. Divide both sides by 2: This isolates x.

    2x / 2 > 10 / 2

    This simplifies to:

    x > 5

  • Equivalent Inequality: x > 5

The resulting inequality, x > 5, is an equivalent inequality to the original inequality, 2x - 3 > 7. They have the same solution set: any number greater than 5 will satisfy both inequalities. For instance, if x is 6:

  • 2(6) - 3 = 9, which is > 7
  • 6 is > 5

Both statements are true, demonstrating the equivalence.

Key Principles for Creating Equivalent Inequalities

  • Addition/Subtraction Property: Adding or subtracting the same number from both sides of an inequality does not change the inequality's solution.
  • Multiplication/Division Property (Positive Number): Multiplying or dividing both sides of an inequality by the same positive number does not change the inequality's solution.
  • Multiplication/Division Property (Negative Number): Multiplying or dividing both sides of an inequality by the same negative number does change the direction of the inequality sign (e.g., > becomes < and vice versa).

Summary

The example from the reference, 2x - 3 > 7 and x > 5, clearly illustrates that equivalent inequalities express the same solution set, despite their different forms. These are useful because they allow us to simplify complex problems, and identify the required values of the unknown variable.

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