Problem 18 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(-27<8 m+5\)

Step-by-Step Solution

Answer

The solution is \(m > -4\), graphed as an open circle at \(-4\) with shading to the right.

Step 1: Isolate the variable term

Start the solution by isolating the term with the variable on one side of the inequality. To do this, subtract 5 from both sides of the inequality: \(-27 - 5 < 8m\).This simplifies to \(-32 < 8m\).

Step 2: Solve for the variable

Next, solve for \(m\) by dividing both sides of the inequality by 8 (the coefficient of \(m\)), as follows: \(\frac{-32}{8} < \frac{8m}{8}\).Simplifying gives \(-4 < m\), which can also be written as \(m > -4\).

Step 3: Graph the solution on a number line

On a number line, represent the solution \(m > -4\) by drawing an open circle at \(-4\) because \(-4\) is not included in the solution set, and shade the number line to the right of \(-4\) to indicate all numbers greater than \(-4\) are included.

Key Concepts

Algebraic Expressions

Algebraic expressions are fundamental components of mathematics that combine numbers, variables, and operations. In the inequality \(-27 < 8m + 5\), we have an algebraic expression on the right-hand side: \(8m + 5\), which consists of a variable term \(8m\) and a constant 5. In an expression, the variable represents an unknown quantity, and it can take various values depending on the context. To handle algebraic expressions effectively:

• Be familiar with basic operations like addition, subtraction, multiplication, and division.
• Understand the role of coefficients, which are numbers multiplying the variable, in this case, the 8 in \(8m\).
• Identify terms clearly: 8m is the variable term, and 5 is a constant term.

Recognizing these components helps in solving inequalities, as it directs which terms need to be manipulated to isolate the variable.

Inequality Graphing

Graphing solutions to inequalities visually represents which values satisfy the inequality. For instance, in the inequality \(m > -4\), our task is to show all possible values of m that are greater than \(-4\). On a number line:

• Use an open circle around \(-4\) to indicate that \(-4\) itself is not part of the solution.
• Shade the region of the number line extending to the right of \(-4\), representing all numbers greater than \(-4\).

These visual cues help understand the solution set and communicate the range of values that meet the inequality. By mastering inequality graphing, you can easily see solutions and check them for correctness.

Variable Isolation

Variable isolation is a crucial step in solving algebraic problems, especially in inequalities. The goal is to get the variable by itself on one side of the inequality. Here’s how it's done:

• Begin by removing any constants from the side containing the variable. For \(-27 < 8m + 5\), subtract 5 from both sides, yielding \(-32 < 8m\).
• Continue by eliminating the coefficient next to the variable. Divide each side by 8, which isolates m, resulting in \(-4 < m\) or \(m > -4\).

By following these steps, you isolate the variable, enabling you to determine the value range for the unknown. This systematic approach ensures clarity and correctness in solving inequalities.