Problem 33 - Algebra 1: Concepts and Skills

Question

Check to see if x = 5 is or is not a solution of the equation or the inequality. $$ 19-2 x>10 $$

Step-by-Step Solution

Answer

No, x = 5 is not a solution to the inequality \(19-2x > 10\).

Step 1: Substitute the given value

Substitute x=5 in the inequality \(19-2x > 10\). It becomes \(19-2(5) > 10\)

Step 2: Simplify the inequality

After simplifying, \(19-10 > 10\) becomes \(9 > 10\)

Step 3: Check the truthfulness of the inequality

The inequality \(9 > 10\) is not true, which shows that x = 5 is not a solution.

Key Concepts

Understanding Algebra and Its Importance

Algebra is a branch of mathematics that uses symbols, typically letters, to represent numbers in equations and inequalities. The main goal of algebra is to determine the value of unknown variables. It provides a way to work with expressions that contain one or more unknowns and helps in forming and solving equations.

• Symbols like 'x', 'y', and 'z' are often used to stand in for unknown values.
• In algebra, you manipulate these symbols to solve for their values.
• Through substitution and simplification, algebra allows us to solve complex equations.

For example, in the inequality given, \[19-2x > 10\],'x' is an unknown variable that we aim to solve for or check a particular value, such as \(x = 5\). By understanding and applying the rules of algebra, you can simplify expressions and solve equations efficiently. It's also crucial for understanding concepts in higher mathematics, sciences, and engineering.

Solution Verification Techniques

Solution verification is crucial in determining whether a proposed value satisfies the given conditions of an equation or inequality. It checks for correctness and ensures that all calculations are accurate.

• First, substitute the proposed value into the equation or inequality.
• Next, simplify the resulting expression to confirm or refute the solution.
• Finally, determine if the simplified expression holds true.

If we take the inequality \[19-2x > 10\] and substitute \(x = 5\), it results in \[19 - 2(5) > 10\].After simplifying to \[9 > 10\],we find that it's false. Thus, this verifies that \(x = 5\) is not a solution. This process not only checks but also reinforces your understanding of how solutions behave in mathematical expressions.

Solving Inequalities

Solving inequalities is similar to solving equations but with key differences. Inequalities express a range of values rather than a single value as solutions. Understanding how to manage these differences is important.

• An inequality compares two expressions and shows the relationship between them.
• The symbols \(>, <, \geq, \leq\) denote the type of comparison.
• The solution set includes all possible values that satisfy the inequality.

In our example, \[19-2x > 10\],solving it would involve the same steps as solving an equation with a focus on maintaining the inequality's direction. It's important to remember that multiplying or dividing both sides by a negative number reverses the inequality sign. Simplification again is key: when we tried substituting \(x = 5\) and got \[9 > 10\],it clearly did not satisfy the inequality, meaning there are other values for \(x\) that do. Understanding inequalities involves practicing how to find these solution sets accurately.