Problem 69 - Algebra 2

Question

Determine whether the given value satisfies the inequality. $$ 4 x^{2}-4 x+1 \leq 10 ; x=2 $$

Step-by-Step Solution

Answer

The value \( x = 2 \) satisfies the inequality.

Step 1: Substitute the Given Value

Substitute the given value of \( x = 2 \) into the inequality \( 4x^2 - 4x + 1 \leq 10 \). This will allow us to determine if this specific value satisfies the inequality.

Step 2: Perform the Calculations

Substitute \( x = 2 \) into the equation: \[ 4(2)^2 - 4(2) + 1 \] Calculate: \[ = 4 \times 4 - 8 + 1 \] \[ = 16 - 8 + 1 \] \[ = 9 \]

Step 3: Compare to the Right Side of the Inequality

Now compare the result from the calculations, \( 9 \), to the right side of the inequality, \( 10 \). Check if \( 9 \leq 10 \).

Step 4: Conclude if the Value Satisfies the Inequality

Since \( 9 \leq 10 \) is true, the value \( x = 2 \) satisfies the inequality \( 4x^2 - 4x + 1 \leq 10 \).

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where the variable is squared, for example, the inequality \( 4x^2 - 4x + 1 \leq 10 \). These types of inequalities are common in algebra and require understanding both algebraic manipulation and comparison. Unlike linear inequalities, which involve straight lines when graphed, quadratic inequalities form parabolic shapes. To solve these, one must determine the values of the variable that make the inequality true.

When solving quadratic inequalities, you seek the range of values for which the inequality holds true. The process often begins by solving the related quadratic equation \( 4x^2 - 4x + 1 = 10 \) to find the critical points. These points help in dividing the number line into intervals that can be tested to verify where the inequality holds true.

In this case, however, you only need to verify if a specific value satisfies the inequality. By substituting the value into the quadratic expression, you check if the resulting statement is true. The concept might seem complex at first, but with practice, it becomes a valuable tool for understanding relationships in algebra.

Algebra

Algebra serves as the foundation for solving quadratic inequalities and many other mathematical problems. It involves using symbols and letters to represent numbers and expressions. By analyzing these symbols, you can formulate relationships and solve problems.

In the given exercise, you use algebra to simplify and evaluate the inequality \( 4x^2 - 4x + 1 \leq 10 \). Here's how to approach it:

• Replace the variable \( x \) with the given value (in this case, 2).
• Perform the arithmetic operations to simplify the expression.
• Compare the result to the number on the right-side of the inequality (10 in this instance).

This exercise highlights the power of algebraic manipulation, as it allows you to make substitutions and follow logical steps to find a solution. Mastering these skills will provide you with a versatile toolkit for tackling a wide range of mathematical dilemmas.

Substitution Method

The substitution method is a straightforward approach often used to solve equations and inequalities, and it plays a crucial role in tackling the problem presented. It involves replacing a variable with a specific value to simplify computations and verify conditions. Let's explore how it works in this context.

1. **Identify the Specific Value:** First, note the given value of the variable you need to check. In our exercise, the value is \( x = 2 \).2. **Substitute into the Expression:** Replace \( x \) in the quadratic expression with 2, resulting in \( 4(2)^2 - 4(2) + 1 \).3. **Simplify the Expression:** Calculate step by step:

• First, evaluate the squared term: \( 4 \times 4 = 16 \).
• Then, carry out the subtraction: \( 16 - 8 = 8 \).
• Finally, add 1 to get 9.

4. **Verify Against the Inequality:** Finally, compare the simplified result (9) to the inequality's constant (10), validating whether \( 9 \leq 10 \) holds true.

The substitution method is particularly useful as it breaks down complex mathematical concepts into digestible parts. By harnessing this method, you can efficiently determine if a specific value satisfies any given inequality.