Problem 56 - Algebra 1

Question

Is the given number a solution of the inequality? $$4 x-1>10 ; 3$$

Step-by-Step Solution

Answer

Yes, '3' is a solution to the inequality \(4 x - 1 > 10\).

Step 1: Substitute the Given Number

To check if '3' is a solution, replace 'x' in the inequality \(4x - 1 > 10\) with '3'. This results in the inequality being \(4*(3) - 1 > 10\)

Step 2: Simplify the Left Hand Side of the Inequality

Calculate \(4*(3) - 1\) which equals to \(12 - 1 = 11\). So the inequality is now \(11 > 10\)

Step 3: Assess the Resulting Inequality

Now, evaluate if 11 is indeed greater than 10. Since it indeed is, that means '3' is indeed a solution present for the inequality \(4 x - 1 > 10\)

Key Concepts

Algebraic Expressions

Algebraic expressions form the cornerstone of many mathematical concepts, including inequalities such as the one in this exercise. An algebraic expression consists of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In our case, the expression is \(4x - 1\). Here, '4x' denotes 4 times the variable \(x\), and '-1' signifies a subtraction operation involved in the expression. Understanding algebraic expressions involves recognizing patterns and simplifying them. In this inequality exercise, we evaluate the expression by substituting a particular value for the variable \(x\) to understand its effect. The objective is to determine whether the resultant expression satisfies the given inequality.

Inequality Solutions

Inequalities express the relationship between algebraic expressions that are not necessarily equal but have either a greater than or less than relationship. In the expression \(4x - 1 > 10\), our goal is to determine if there are any values of \(x\) that make the left side greater than 10. Finding solutions to inequalities means looking for values of \(x\) that maintain the truth of these relationships.

Sometimes, inequalities have endless solutions. In other cases, they have a limited set of solutions. In the current problem, we substitute a specific value to check if it makes the inequality true. This requires simplifying the expression for a given \(x\) and comparing it to the number on the right side of the inequality.

Substitution Method

The substitution method helps solve or check algebraic equations and inequalities by replacing variables with specific numbers. It's a straightforward yet powerful tool. In this scenario, we are given a specific value (in this case, '3'). By substituting '3' for \(x\) in \(4x - 1 > 10\), we replace the abstract expression with actual numbers, simplifying it to \(4 * 3 - 1\).

This encourages students to compute and verify the truth of the inequality. A crucial part of understanding the substitution method involves carrying out arithmetic accurately and making clear comparisons. Observing whether the process results in an inequality that holds (such as \(11 > 10\)) confirms that the selected number satisfies the inequality condition.