Problem 33 - Algebra 1: Concepts and Skills

Question

Check to see if the given value of the variable is or is not a solution of the inequality. $$ 6 x<35 ; x=6 $$

Step-by-Step Solution

Answer

No, x = 6 is not a solution to the inequality \(6x < 35\).

Step 1: Substitute the given value

Replace the variable 'x' in the inequality with the given value 6. This leads to the inequality: \(6 * 6 < 35\). Calculating the left-hand side, we get the resultant inequality: \(36 < 35\).

Step 2: Evaluate the inequality

Check whether the inequality holds true. In this case, we observe that 36 is not less than 35. Hence, the inequality does not hold true.

Step 3: Conclusion

Since the resultant inequality \(36 < 35\) does not hold true, we conclude that x = 6 is not a solution to the inequality \(6x < 35\).

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving both equations and inequalities. Here's how it works:

• Identify the variable: First, figure out which variable needs a value. For inequalities, this is the variable you're checking against the inequality.
• Substitute the given value: Replace the variable in the inequality with the given number. In our exercise, we substitute 6 for the variable 'x' in the inequality: \(6x < 35\).
• Simplify: Perform the arithmetic operations to simplify the expression down to a basic inequality without variables. This helps us check if the inequality holds.

The substitution method essentially transforms an inequality with variables into a straightforward numeric comparison. It's a critical step in determining if a given number is a solution to the inequality. This process allows you to focus fully on comparing simple numeric values instead of managing algebraic forms.

Algebraic Inequality

Algebraic inequalities express a range of possible solutions rather than a single solution, unlike equations. They bring the element of comparison into solving, which makes them quite interesting. Here's more about them:

• Format: Inequalities usually involve expressions on both sides of a comparison symbol (like <, >, ≤, or ≥). For example, \(6x < 35\) implies that the expression \(6x\) must be less than \(35\).
• Operations: You can perform similar operations on inequalities as on equations, such as addition, subtraction, multiplication, or division. In our example, multiplying \(x\) by 6 is one such operation, which gave us \(36\) when \(x = 6\).
• Range of Solutions: An inequality often represents a continuum of numbers. Therefore, a single value might not satisfy the inequality, pointing out it isn't the correct solution.

Handling inequalities requires careful comparison to ensure the correctness of signs and ranges. They are indispensable in practical scenarios where you’re not seeking an exact solution but a feasible range.

Solution Verification

Verifying a solution is the final and crucial step in confirming whether a proposed value satisfies an inequality. Let's delve in:

• Substitute and Simplify: Initially, substitute the proposed value into the inequality and perform the arithmetic to check the resultant inequality. For instance, substituting \(x = 6\) into \(6x < 35\) gave us \(36 < 35\).
• Compare: Examine the simplified inequality. For our instance, the statement \(36 < 35\) is false, meaning the selected value did not fulfill the inequality.
• Conclusion: Based on this comparison, conclude whether the proposed value is a solution. When the statement is false, like in our example, \(x = 6\) is not a solution to \(6x < 35\).

This step is all about assuring accuracy, ensuring that you have correctly verified whether a value meets the inequality's requirement. It's vital to confirm if a value aligns with the inequality, strengthening your understanding and confidence in algebraic processes.