Problem 68 - Algebra 2

Question

Determine whether the given value satisfies the inequality. $$ 4 x^{2}+2 x-3 \geq 0 ; x=-1 $$

Step-by-Step Solution

Answer

The value \( x = -1 \) does not satisfy the inequality.

Step 1: Substitute the given value into the inequality

Start by substituting the given value of \( x = -1 \) into the inequality \( 4x^2 + 2x - 3 \geq 0 \). This means we replace \( x \) with \( -1 \) in each instance.

Step 2: Calculate the expression

After substitution, the inequality becomes \( 4(-1)^2 + 2(-1) - 3 \geq 0 \). Calculate each part: - \((-1)^2 = 1\), so \(4 \times 1 = 4\).- \(2(-1) = -2\).Now combine these: \(4 - 2 - 3\).

Step 3: Simplify the expression

Add and subtract the numbers to simplify the expression: - \(4 - 2 = 2\)- \(2 - 3 = -1\)So, the entire expression simplifies to \(-1\).

Step 4: Check the inequality

Now determine if \(-1 \geq 0\). Since \(-1\) is not greater than or equal to zero, the inequality is not satisfied.

Key Concepts

Inequality Solving

Solving inequalities is all about finding out when a particular statement is true or false. In mathematical terms, an inequality is a statement about the relative size or order of two expressions, typically involving the symbols \( >, <, \geq, \) or \( \leq \). To determine whether a given number satisfies an inequality, we substitute the number into the inequality and check if the resulting statement holds true.In our exercise, the inequality is \( 4x^2 + 2x - 3 \geq 0 \), and we want to know if \( x = -1 \) makes this statement true. The initial step is to substitute \( x = -1 \) into the inequality. If after substitution the inequality evaluates to true (for example, it becomes \( 1 \geq 0 \)), the given value satisfies the inequality. Otherwise, it doesn't.Key steps include:

• Substitute the value into the inequality.
• Calculate the resulting expression.
• Determine whether the inequality is satisfied with the result.

By following these steps, we can solve inequalities by interpreting and manipulating expressions to analyze their truthfulness.

Substitution Method

The substitution method is a powerful technique used to simplify complex mathematical expressions by replacing a variable with a specific value. This technique helps us to evaluate expressions more easily and quickly check the truthfulness of equations and inequalities. For our exercise, we substitute \( x = -1 \) into the expression \( 4x^2 + 2x - 3 \geq 0 \). This involves replacing every occurrence of \( x \) with \( -1 \). This simplifies the computation process and allows us to focus on simple arithmetic operations thereafter.Steps for substitution:

• Identify the variable to be replaced and its value.
• Replace the variable in the expression with its given value.
• Proceed with simplifying and solving the expression.

Once substitution is applied, the problem becomes straightforward arithmetic, which can be systematically solved to check the inequality.

Simplifying Expressions

Simplifying expressions is critical when solving quadratic inequalities. By breaking down the expression, we turn complex mathematical problems into manageable calculations. It involves performing arithmetic operations and combining like terms to reach a simpler form that is easier to analyze.In solving \( 4x^2 + 2x - 3 \geq 0 \) for \( x = -1 \), simplifying involves calculating \( 4(-1)^2 + 2(-1) - 3 \). Handle each term separately:

• First, calculate \( (-1)^2 = 1 \), making the term \( 4 \times 1 = 4 \).
• Next, find \( 2 \times -1 = -2 \), which simplifies the expression to \( 4 - 2 \).
• Finally, subtract 3: \( 2 - 3 = -1 \).

This simplification helps determine whether \( -1 \geq 0 \) is true or false. Here, the simplified result \( -1 \) is not greater than or equal to zero, leading us to conclude that the inequality is not satisfied. Simplifying expressions lets us see results clearly and ensures accurate conclusions.