Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each value of \(x\) is a solution of the inequality. Inequality \(5 x-12>0\) Values (a) \(x=3\) (b) \(x=-3\) (c) \(x=\frac{5}{2}\) (d) \(x=\frac{3}{2}\)

Step-by-Step Solution

Answer

Values of \( x \) that satisfy the inequality \( 5x - 12 > 0 \) are \( x = 3 \) and \( x = 5/2 \).

Step 1 - Substituting \( x = 3 \)

Substitute \( x \) with 3 into the given inequality. This will give us \( 5*3 - 12 > 0 \), which simplifies to \( 15 - 12 > 0 \). Thus, \( 3 > 0 \). So, \( x = 3 \) is a solution.

Step 2 - Substituting \( x = -3 \)

Substitute \( x \) with -3 into the given inequality. This will give us \( 5*-3 - 12 > 0 \), which simplifies to \( -15 - 12 > 0 \). Thus, \( -27 > 0 \). So, \( x = -3 \) is not a solution.

Step 3 - Substituting \( x = 5/2 \)

Substitute \( x \) with \( 5/2 \) into the given inequality. This will give us \( 5*(5/2) - 12 > 0 \), which simplifies to \( 25/2 - 12 > 0 \). Thus, \( 0.5 > 0 \). So, \( x = 5/2 \) is a solution.

Step 4 - Substituting \( x = 3/2 \)

Substitute \( x \) with \( 3/2 \) into the given inequality. This will give us \( 5*(3/2) - 12 > 0 \), which simplifies to \( 15/2 - 12 > 0 \). Thus, \( -1.5 > 0 \). So, \( x = 3/2 \) is not a solution.

Key Concepts

Algebra

Algebra is like the language of mathematics. It uses letters and symbols to represent numbers and express relations between them. This helps to solve equations and inequalities by finding the values of unknown variables that make an expression true. In this exercise, we look at an inequality: \(5x - 12 > 0\). Here, you are working to determine if different values of \(x\) make this statement valid. Think of algebra as a balancing act, where both sides of an equation or inequality need to be equal or maintain the inequality respectively. When dealing with inequalities like \(5x - 12 > 0\), algebra allows you to manipulate the expression to find the solution. By isolating \(x\), you gain insights into which values make the inequality true. You achieve this by performing similar operations on both sides of the inequality, which maintains its balance and doesn't alter the relationship. This understanding is fundamental in tackling various math problems!

Solution Verification

Solution verification is a critical step to confirm whether a proposed solution actually satisfies the given inequality. It involves substituting values into the inequality and checking if the result holds true. For example, in the problem, you were asked to verify the solutions for different values:

• Substitute \(x = 3\) into the inequality \(5x - 12 > 0\) and check if the statement is true.
• Perform the same for \(x = -3\), \(x = \frac{5}{2}\), and \(x = \frac{3}{2}\).

When you verify these solutions, you can confidently conclude which values of \(x\) satisfy the inequality. This step is vital to prevent errors and ensure accuracy in your math work. It's like double-checking your answers in a math quiz to make sure everything is correct!

Substitution Method

The substitution method is a straightforward technique used to simplify the process of evaluating expressions within equations and inequalities. By substituting specific values in place of the variable, you determine whether these values satisfy the equation or inequality.In the context of this exercise, substitution helps determine if different \(x\) values solve the inequality \(5x - 12 > 0\). Here's how you can efficiently apply the substitution method:

• Take the value of \(x\) you want to test and substitute it into the inequality.
• Simplify the expression to check if the inequality holds true.
• Repeat for each value provided in the problem.

Using substitution, you systematically understand each potential solution without needing to rework the inequality from scratch. By substituting the given values, you effectively evaluate their validity, turning an abstract equation into a simple arithmetic problem.