Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$4 p-1 \geq 8 ; 2$$

Short answer

No, 2 is not a solution of the inequality \(4p - 1 \geq 8\).

Step-by-step solution

Understand the Problem

The inequality to check is \(4p - 1 \geq 8\). The number to check is \(2\). It means that we need to substitute \(p = 2\) into the inequality and simplify the expression to check if it satisfies the inequality.

Substitute Number

Substitute \(p = 2\) into the inequality, giving us (4*2) - 1 which simplifies to \(7\).

Check the Inequality

We need to check if \(7 \geq 8\). The inequality \(7 \geq 8\) is not true, so 2 is not a solution of the inequality \(4p - 1 \geq 8\).

Key concepts

Inequality Substitution

Understanding the concept of inequality substitution is crucial when solving algebraic problems. It involves replacing a variable with a specific value to determine whether the resulting statement holds true. In the given exercise, the inequality is \(4p - 1 \geq 8\), and we need to investigate if the number 2 is a solution.

Substitute \(p = 2\) into the inequality to get \(4 \times 2 - 1\), which simplifies to 7. This process is the essence of inequality substitution: taking an unknown quantity and replacing it with a known value to draw conclusions about the relationship of the numbers.

Always double-check your substitution to ensure accuracy. Also, apply the correct order of operations, sometimes referred to as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Algebraic Inequalities

Algebraic inequalities, such as the one presented in the exercise \(4p - 1 \geq 8\), represent a broad range of mathematical scenarios where two expressions may not be equal, but there exists a relation of 'greater than', 'less than', or their 'or equal to' counterparts.

Inequalities symbolize a range or set of possible values rather than a single solution like equations. Understanding the properties of inequalities is fundamental; for instance, adding or subtracting the same number from both sides does not change the inequality's direction, whereas multiplying or dividing by a negative number reverses the inequality's direction.

When checking solutions for inequalities, it is crucial to remember that you are testing if the values satisfy the conditions set by the inequality, not trying to find an exact match as in an equation.

Inequality Simplification

The process of inequality simplification involves minimizing the inequality to its simplest form to make the comparison between two expressions clearer. Simplification may include combining like terms, distributing multiplication over addition, or canceling out numbers on both sides.

In the solution provided, simplification occurs after substitution, reducing \(4 \times 2 - 1\) to 7. Once simplified, the inequality \(7 \geq 8\) is easy to assess. Despite the simplicity of this scenario, remember that some inequalities include more complex simplifications, which can include factoring expressions or dealing with fractions.

Simplifying each step as far as possible before moving on to the next can prevent errors and provide a clearer path to determining whether the value in question is indeed a solution to the inequality.