Problem 77 - Elementary Algebra Within Reach

Question

Determine whether the statement is true or false. Justify your answer. The inequality \(x+6>0\) is equivalent to \(x>-6\).

Step-by-Step Solution

Answer

The statement is true; the inequality \(x+6>0\) is equivalent to \(x>-6\).

Step 1: Analyze the Original Inequality

To start off, take a look at the original inequality: \(x+6>0\). The goal is to isolate the variable \(x\), to see if it matches the proposed equivalent.

Step 2: Solve the Original Inequality

To isolate \(x\), subtract \(6\) from both sides of the inequality \(x+6>0\). This results in \(x>-6\).

Step 3: Compare the Transformed Inequality with the Proposed One

The transformed inequality \(x>-6\) is exactly the same as the proposed equivalent from the task. Therefore, the statement stating that the inequalities are equivalent is true.

Key Concepts

Solving Inequalities

In algebra, solving inequalities is a fundamental skill that often mirrors solving equations, with a couple of key differences. For an inequality like x+6>0, you're looking for all values of x that make the inequality true, not just a single solution. The process involves isolating the variable on one side to make a statement about its possible values.

To solve the given inequality, start by subtracting 6 from both sides of the inequality, effectively peeling away the added number to reveal the variable's range of values. The resulting inequality, x>-6, tells us that any number greater than -6 is a solution to the initial inequality.

• Keep the inequality sign pointing the same way unless you multiply or divide by a negative number (in which case, flip the sign).
• When the variable is isolated, make sure to read the inequality correctly: x>-6 states that x is any number greater than -6, not including -6 itself.

Algebraic Reasoning

Algebraic reasoning allows you to approach problems systematically by identifying relationships between elements and applying operations to both sides of an equation or inequality without changing the truth of the statement. When given the inequality x+6>0, algebraic reasoning prompts you to recognize that subtracting 6 from both sides will not change the inequality's truth status.

Algebraic reasoning extends to understanding the properties of inequalities and how they behave differently from equations. For instance, knowing that adding or subtracting the same number to both sides of an inequality doesn't affect the solution set is key when solving the given problem. It's also about understanding when and why you need to switch the direction of the inequality sign, which is not necessary in this exercise since we are only dealing with addition and subtraction.

Equivalent Expressions

Equivalent expressions in algebra are different expressions that represent the same relationship or value. For instance, the initial inequality x + 6 > 0 and the simplified expression x > -6 are equivalent because they both represent the same set of numbers that satisfy the inequality.

Identifying equivalent expressions requires an understanding of how algebraic operations affect an expression. In the provided exercise, subtracting 6 from both sides maintains the inequality's value, meaning the transformation does not disrupt the relationship between the numbers.

• 'Equivalent' does not mean 'identical' but rather 'having the same value or effect'.
• To test for equivalency, you can pick numbers and see if they satisfy both expressions, or prove it through algebraic manipulation, as shown in the solution.