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what values of b satisfy 3 2b 3 2 36

what values of b satisfy 3 2b 3 2 36

less than a minute read 05-02-2025
what values of b satisfy 3 2b 3 2 36

This article will guide you through solving the inequality 3 ≤ 2b + 3 ≤ 23 to find the values of 'b' that satisfy the condition. We'll break down the process step-by-step, making it easy to understand.

Understanding Compound Inequalities

The given expression, 3 ≤ 2b + 3 ≤ 23, is a compound inequality. This means it combines two inequalities (3 ≤ 2b + 3 and 2b + 3 ≤ 23) into a single statement. To solve it, we need to isolate 'b' in the middle.

Solving the Inequality: Step-by-Step

  1. Subtract 3 from all parts: To begin isolating 'b', we subtract 3 from each part of the inequality:

    3 - 3 ≤ 2b + 3 - 3 ≤ 23 - 3

    This simplifies to:

    0 ≤ 2b ≤ 20

  2. Divide all parts by 2: The next step is to divide all parts of the inequality by 2 to solve for 'b':

    0/2 ≤ 2b/2 ≤ 20/2

    This simplifies to:

    0 ≤ b ≤ 10

The Solution: Values of 'b'

Therefore, the values of 'b' that satisfy the inequality 3 ≤ 2b + 3 ≤ 23 are all numbers between 0 and 10, inclusive. This can be expressed in interval notation as [0, 10]. This means 'b' can be any number greater than or equal to 0 and less than or equal to 10.

Checking the Solution

Let's test a few values within the solution range to verify:

  • b = 0: 3 ≤ 2(0) + 3 ≤ 23 => 3 ≤ 3 ≤ 23 (True)
  • b = 5: 3 ≤ 2(5) + 3 ≤ 23 => 3 ≤ 13 ≤ 23 (True)
  • b = 10: 3 ≤ 2(10) + 3 ≤ 23 => 3 ≤ 23 ≤ 23 (True)

Now let's test a value outside the solution range:

  • b = 11: 3 ≤ 2(11) + 3 ≤ 23 => 3 ≤ 25 ≤ 23 (False)

These checks confirm our solution is correct.

Conclusion

The inequality 3 ≤ 2b + 3 ≤ 23 is satisfied by all values of 'b' within the interval [0, 10]. This means 'b' can be any real number between 0 and 10, including 0 and 10 themselves. Understanding compound inequalities and following the steps for solving them is crucial in algebra and related fields.

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