Problem 53 - Algebra 1: Concepts and Skills

Question

Check whether the given value of the variable is a solution of the inequality. (Lesson 1.4) $$ 7 y+6>10 ; y=3 $$

Step-by-Step Solution

Answer

The given value of y, which is 3, is indeed a solution to the inequality '7y + 6 > 10'.

Step 1: Substitute value of y into the inequality

Initially, replace 'y' in the inequality '7y + 6 > 10' with '3'. This gives '7*3 + 6 > 10', which simplifies to '21 + 6 > 10'.

Step 2: Simplify the left side of the inequality

Carry out the arithmetic operation on the left-hand side of inequality, resulting in '27 > 10'.

Step 3: Verify the inequality

The inequality '27 > 10' holds true as 27 is greater than 10, confirming that the provided value for 'y', which is '3', is indeed a solution to the original inequality.

Key Concepts

Substitution

Substitution is an essential technique in solving algebraic problems, such as inequalities. It involves replacing a variable with a given numerical value to test whether an expression holds true. In the case of the exercise, we start by substituting the value of the variable \( y \) with 3 into the given inequality, \( 7y + 6 > 10 \). This process transforms the inequality into a form that can be easily simplified and evaluated. By substituting \( y = 3 \), the expression changes to \( 7 \, \times \, 3 + 6 > 10 \). Substitution is key because it allows us to directly check the validity of the expression by reducing the problem from a variable-based expression to a simple numerical comparison.

Arithmetic Operations

Arithmetic operations are the basic calculations we perform with numbers, such as addition, subtraction, multiplication, and division. These operations allow us to simplify and solve expressions and inequalities.

For our specific inequality, \( 7 \, \times \, 3 + 6 > 10 \), performing the multiplication first results in \( 21 \). Then, adding \( 6 \) to \( 21 \) gives \( 27 \). Understanding the correct order of arithmetic operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), is crucial in simplifying expressions correctly.

These operations help confirm whether the simplified form \( 27 > 10 \) is true or false, by comparing the result with the other side of the inequality.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. Variables in algebraic expressions represent unknown values and can be replaced with specific numbers, like in this exercise.

The original inequality \( 7y + 6 > 10 \) is an example of an algebraic expression where \( y \) is the variable. Understanding how to manipulate algebraic expressions is a fundamental part of solving inequalities. Through substitution and arithmetic operations, these expressions can be transformed into simpler forms that can be evaluated easily.

• Make sure to identify variables correctly.
• Replace variables with their given values when instructed.
• Carry out arithmetic operations to simplify the expressions.

These steps help students recognize patterns and relationships in algebra, enabling them to solve more complex mathematical problems in the future.