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Chapter 6: Problem 48

Check whether the given number is a solution of the inequality. $$2 z-3<4 ; 4$$

Short Answer

Expert verified
The number 4 is not a solution to the inequality \(2z - 3 < 4\) because the substituting 4 into the inequality results in an invalid statement.

Step by step solution

01

Identify the Proposed Solution

This exercise provides a number: 4, as a proposed solution to the inequality. This means 4 is suggested as a potential value for \(z\).
02

Substitute the Proposed Solution

Replace \(z\) with 4 in the inequality \(2z - 3 < 4\). Which gives \(2 (4) - 3 < 4\), simplifying it gives 5 < 4.
03

Check the Validity of the Inequality

Examine whether the final statement 5 < 4 is valid or not. Since 5 is not less than 4, the statement is false.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

The Substitution Method
The substitution method is a straightforward approach used to determine if a given number is a solution to an inequality or equation. It involves replacing the variable in the inequality with the proposed value to see if the resulting expression holds true.
In this exercise, the variable is \( z \), and the problem suggests the potential solution is 4. Thus, the substitution method requires us to replace \( z \) with 4 in the inequality \( 2z - 3 < 4 \).
This substitution leads to the expression \( 2(4) - 3 < 4 \). By performing the arithmetic, we simplify it to 5 < 4. This method helps us create a clear path to check whether our proposed solution satisfies the inequality.
Understanding Validity Check
After substitution, it's important to check the validity of the resulting expression. This step assesses whether the simplified inequality makes logical sense.
From our previous substitution, we arrived at 5 < 4. In a validity check, we determine if this statement is true.
Since 5 is not less than 4, the statement is false. Hence, this proposed solution does not satisfy the inequality.
  • Validity checks are crucial to confirm the correctness of potential solutions.
  • These checks prevent incorrect solutions from being accepted.
By performing a validity check, learners gain confidence that they are validating their solutions properly.
Assessing the Proposed Solution
In any problem involving inequalities, it's essential to assess the proposed solution's feasibility. Here, the proposed solution was 4, but through our substitution and validity check, we found it did not satisfy the inequality.
So, what does this mean? The proposed solution is rejected as a valid answer for the inequality \(2z - 3 < 4 \).
If a proposed solution does not work, it directly guides us to understand that either another solution must be tested, or no solution exists if additional constraints are given.
  • Always double-check substitutions and simplifications.
  • A rejected solution can inform further problem-solving strategies.
Being methodical with each proposed solution ensures a more comprehensive understanding of the inequality and the variable鈥檚 possible values.

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