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Chapter 1: Problem 3

Determine whether or not the given value is a solution. 3y?4=5;???

Short Answer

Expert verified
No, \(x = 1\) is not a solution as it makes the inequality \(-1 < -5\), which is false.

Step by step solution

01

Substitute the given value into the inequality

The given inequality is \(2x - 3 < -5\) and the value to check is \(x = 1\). Substitute \(x = 1\) into the inequality to get \(2(1) - 3 < -5\).
02

Simplify the left side of the inequality

Simplify the expression \(2(1) - 3\) by performing the multiplication first: \(2 \cdot 1 = 2\). So the inequality becomes \(2 - 3 < -5\).
03

Perform the subtraction

Subtract 3 from 2 to simplify the expression: \(2 - 3 = -1\). Hence, we have \(-1 < -5\).
04

Evaluate the inequality

Compare \(-1 < -5\) to see if it holds true. Since \(-1\) is not less than \(-5\), the inequality is false.

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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 inequality is a critical step in understanding inequalities. In this context, solution verification means checking to see if a specific value satisfies the inequality condition.
  • Start by substituting the value into the inequality equation.
  • If the inequality holds true after the substitution, the value is a solution.
  • If the inequality does not hold, then the value is not a solution.
It's essential to verify solutions to ensure accuracy in mathematical reasoning and to confirm that the logic applied is correct. Mistakes in this step could lead to incorrect conclusions.
Substitution Method
The substitution method involves replacing the variable in an inequality with a given number to see if the inequality condition is met. This is a basic but crucial skill in solving inequalities and other algebraic expressions.
  • Identify the variable in the inequality and note the value that needs to be checked.
  • Substitute the identified value directly into the position of the variable in the inequality.
For instance, in the inequality \(2x - 3 < -5\), we substitute \(x = 1\). This gives us \(2(1) - 3 < -5\), which sets the stage for further simplification.
Simplification Process
After substitution, simplifying the inequality is the next essential step. Simplification means performing the arithmetic operations required to transform the expression into a simpler form.
  • Follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
  • In the example \(2(1) - 3 < -5\), start with the multiplication \(2 \cdot 1 = 2\).
  • Next, perform the subtraction: \(2 - 3\), which results in \(-1\).
The simplification will make it clearer whether or not the inequality condition is fulfilled.
Inequality Evaluation
Once the inequality is simplified, evaluate the outcome to determine if the initial value is indeed a solution. This involves comparing numerical values based on the inequality symbols.
  • In the example, we evaluate \(-1 < -5\).
  • A careful assessment shows that \(-1\) is not less than \(-5\).
  • This leads us to conclude that the initial assumption about the value being a solution is incorrect.
Understanding how to evaluate inequalities accurately ensures a better grasp of the concepts involved and prevents potential errors in future calculations.

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