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Chapter 1: Problem 2

Determine whether or not the given value is a solution. 4x?3=?7;???x=?1

Short Answer

Expert verified
Yes, \( x = 1 \) is a solution.

Step by step solution

01

Substitute the Given Value

First, substitute the given value of \( x = 1 \) into the inequality \( -3x + 1 > -10 \). This means replacing every occurrence of \( x \) in the inequality with \( 1 \).
02

Simplify the Left Side

After substitution, the inequality becomes \( -3(1) + 1 > -10 \). Calculate \(-3 \times 1\), which gives \(-3\). Now add 1 to this result: \(-3 + 1 = -2\).
03

Compare Both Sides of the Inequality

The inequality simplifies to \(-2 > -10\). Compare the numbers: \(-2\) is indeed greater than \(-10\).
04

Conclusion

Since \(-2 > -10\) holds true, the given value \( x = 1 \) satisfies the inequality \( -3x + 1 > -10 \). Thus, \( x = 1 \) is a solution to the inequality.

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Key Concepts

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

Solution Verification
Verifying whether a given value is a solution to an algebraic inequality is essential. To do this, we replace the variable with the provided value and check if the inequality holds true. This process ensures that when the variable is substituted, the inequality remains valid.
In the given exercise, our task was to determine if \( x = 1 \) is a solution to the inequality \(-3x + 1 > -10\).
  • Start by substituting \( x = 1 \) into the inequality.
  • Then, simplify the expression.
  • Finally, compare the resulting values on each side of the inequality.
This sequence provides clarity on the legitimacy of the proposed solution.
Algebraic Substitution
Algebraic substitution is a pivotal technique in solving inequalities. It involves replacing a variable with a specific value to simplify the expression or equation.
In our example, substituting \( x = 1 \) into the inequality \(-3x + 1 > -10\) transforms it into a numeric comparison.
  • Replace every appearance of the variable \( x \) with the given value.
  • Perform standard arithmetic operations to simplify the expression.
By doing so, you convert the abstract inequality into a more tangible numeric form, making it easier to evaluate and understand.
Algebraic Inequalities
Algebraic inequalities express a relationship between two expressions where one is greater than, less than, or sometimes equal to the other. They are similar to equations but incorporate inequality signs like \(>\), \(<\), \(\geq\), and \(\leq\) instead of an equals sign.
In the case of \(-3x + 1 > -10\), it indicates that the expression on the left is greater than what is on the right for certain values of \( x \).
  • This exercise shows that the inequality holds true for \( x = 1 \).
  • Understanding the intent of inequalities helps in finding valid solutions.
Unlike equations, inequalities often have a range of solutions, not just a single answer.

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