Problem 29 - Algebra 1: Concepts and Skills

Question

Check to see if x = 5 is or is not a solution of the equation or the inequality. $$ 2.5>1.2 x-3 $$

Step-by-Step Solution

Answer

No, x = 5 is not a solution of the inequality \(2.5 > 1.2x - 3\)

Step 1: Substitution

Substitute x=5 into the inequality which gives us \(2.5 > 1.2(5) - 3 \)

Step 2: Simplification

Simplify the right hand side of the inequality to give \(2.5 > 6 - 3\) which further simplifies to \(2.5 > 3\)

Step 3: Inequality Verification

The inequality \(2.5 > 3\) is not true, hence x = 5 is not a solution of the inequality.

Key Concepts

Substitution Method

The substitution method is a crucial technique in algebra, particularly useful for solving equations and inequalities. The primary idea is to replace a variable with a given number to determine if it satisfies the equation or inequality.
For example, if we have an inequality such as \(2.5 > 1.2x - 3\) and we're asked to check if \(x = 5\) is a solution:

• Start by substituting \(x\) in the inequality with \(5\).
• This gives you \(2.5 > 1.2(5) - 3\).

By substituting, we are effectively testing the correctness of the inequality for a particular value of \(x\). This is an essential step before moving on to solve or verify the inequality.

Inequality Verification

Inequality verification requires analyzing the transformed inequality to see if it holds true. This involves evaluating the inequality after substituting the variable, and then confirming whether the resulting statement is valid.
Continuing from the substitution:

• The inequality now is \(2.5 > 1.2(5) - 3\), which simplifies down to \(2.5 > 3\).
• It's important to carefully check whether \(2.5\) is indeed greater than \(3\).

Since \(2.5\) is not greater than \(3\), the inequality does not hold true, which means \(x = 5\) is not a solution to the inequality. Understanding how to verify inequalities is a vital skill, ensuring you can accurately determine the truthfulness of statements involving inequalities.

Simplification in Algebra

Simplification in algebra involves reducing expressions to their simplest form, thus making them easier to work with. Simplifying is an indispensable step when solving equations or inequalities, as it allows for straightforward computation and verification.
In our example:

• After substituting, we had \(2.5 > 1.2(5) - 3\).
• Calculating further, we simplify \(1.2(5)\) to \(6\), resulting in the simplified inequality: \(2.5 > 6 - 3\).
• Then, compute \(6 - 3\) which yields \(3\). So, we have \(2.5 > 3\).

Simplification ensures that complex expressions are reduced to a form that is easier to interpret and verify. This process is pivotal, especially in academic and problem-solving settings, where precise calculations and clear conclusions are necessary.