Problem 70 - Algebra 2

Question

Determine whether the given value satisfies the inequality. $$ 6 x^{2}+3 x > 8 ; x=0 $$

Step-by-Step Solution

Answer

No, the value \( x = 0 \) does not satisfy the inequality.

Step 1: Substitute the Value of x

First, replace the variable \( x \) in the inequality with the given value, which is \( x = 0 \). This gives us: \[ 6(0)^2 + 3(0) > 8 \]

Step 2: Simplify the Equation

Calculate the terms after substitution. \[ 6(0)^2 = 0 \] and \[ 3(0) = 0 \], so \[ 0 + 0 > 8 \] simplifies to \[ 0 > 8 \].

Step 3: Evaluate the Inequality

Determine if the inequality is true. We have \( 0 > 8 \), which is not true because 0 is not greater than 8.

Key Concepts

Substitution

Substitution is an essential technique when working with inequalities and equations. In this context, substitution involves replacing the variable in an expression with a given value. This helps to simplify the expression or inequality so that it can be more easily evaluated. For example, in the given exercise, we substitute the value of \( x \) with 0 in the inequality \( 6x^2 + 3x > 8 \). To do this, every occurrence of \( x \) is replaced with 0, leading to a new expression: \( 6(0)^2 + 3(0) > 8 \). This substitution step is key because it transforms the inequality into a form that can be directly evaluated, facilitating the next steps in problem-solving.Substitution is a straightforward process, but it's important to perform it correctly to avoid errors in further calculations. Double-checking at this stage can prevent simple mistakes from growing into larger issues later on.

Simplification

Simplification goes hand-in-hand with substitution when dealing with expressions and inequalities. After substituting the variable with a specific value, simplification aims to reduce the complexity of the expression by performing basic arithmetic operations. In the original exercise, after substituting \( x = 0 \), the expression \( 6(0)^2 + 3(0) > 8 \) becomes a much simpler form, \( 0 + 0 > 8 \), which is equal to just \( 0 > 8 \). This step involves breaking down the expression into its simplest components:

• Calculate powers: \( 6(0)^2 = 0 \).
• Multiply: \( 3(0) = 0 \).
• Combine and simplify: Add the results to form \( 0 + 0 = 0 \).

By simplifying, you convert an expression into an easily understandable and evaluable form. Make sure to perform each arithmetic operation carefully to ensure the accuracy of the final result.

Evaluating Inequalities

Once an inequality is simplified, the next step is to evaluate it. Evaluating an inequality means determining whether the statement is true or false. In other words, we check if the inequality holds with the given substitution.For the problem at hand, evaluating the simplified inequality \( 0 > 8 \) shows that it doesn't hold true because zero is not greater than eight. Understanding this step is crucial because it confirms whether the proposed solution or substitution meets the inequality's condition.Evaluating involves:

• Comparing both sides of the inequality after simplification.
• Deciding whether the left side of the inequality is indeed greater (or smaller) than the right side, as required by the inequality operator (>, <, \( \ge \), or \( \le \)).

In summary, this step answers the question: Does the substitution satisfy or violate the original inequality?