Problem 20 - Elementary Algebra

Question

Graph the linear equations and inequalities. $$ 3 m-7 \leq 8 $$

Step-by-Step Solution

Answer

Answer: The area below the horizontal line at $m = 5$ satisfies the inequality.

Step 1: Solve the inequality for m

First, we need to isolate m on one side of the inequality:$$ 3m - 7 \leq 8 $$Add 7 to both sides:$$ 3m \leq 15 $$Now, divide both sides by 3:$$ m \leq 5 $$The inequality for m is:$ m \leq 5 \(

Step 2: Graph the inequality

Now, let's graph the inequality. Since it's in the form of \)m \leq 5\(, we can treat it as a function of \)x\(, such as:\) m(x) \leq 5 $$Since the inequality is "\(\leq\)", we'll use a solid line to represent the line \(m(x) = 5\). 1. Draw a solid, horizontal line at \(m = 5\). 2. The inequality is \(m \leq 5\), so we'll shade the area below the line to represent all the points where \(m \leq 5\). The graph of the linear inequality.$$ 3m - 7 \leq 8 $$Is a horizontal line at \(m = 5\), with the region below the line shaded.

Key Concepts

Graphing Inequalities

A crucial way to understand inequalities is through their graphical representation. For the inequality \(3m - 7 \leq 8\), graphing helps visualize the solution set. After solving, we get \(m \leq 5\). Here's how to graph it:

• Draw a horizontal line at \(m = 5\). This is because \(m\) acts similarly to \(y\) in a coordinate system.
• Since it’s a "less than or equal to" (\(\leq \)) inequality, use a solid line to indicate that the line itself is included in the solution.
• Shade the region below the line. The shaded area includes all points where \(m\) is less than or equal to 5, showing all possibilities that satisfy the inequality.

Graphing can make solving inequalities clearer, particularly when dealing with multiple conditions or variables. You see the possible solutions visually, helping you understand and check the results.

Solving Inequalities

Solving inequalities is similar to solving equations, but with special rules. For the inequality \(3m - 7 \leq 8\), start by isolating the variable on one side.

• Add 7 to both sides: \(3m \leq 15\). This step removes constants attached to \(m\).
• Next, divide both sides by 3: \(m \leq 5\). This isolates \(m\), giving us the solution.

When solving inequalities, remember that multiplying or dividing by a negative number reverses the inequality. Keep this in mind to ensure accurate results. Always check your solution by substituting it back into the original inequality to see if it holds true. This helps you confirm that the solution set is correct.

Linear Equations

Linear equations are expressions where variables hold constant first-degree powers. For example, in the task \(3m - 7 \leq 8\), the expression \(3m - 7\) resembles a linear equation.

• To visualize or solve a linear equation, you first manipulate terms to isolate the variable, much like solving an inequality.
• Graphically, linear equations produce straight lines. Each point on the line corresponds to a solution of the equation.

Linear equations form the foundations of solving inequalities because often, the first step is rewriting the inequality as a linear equation by treating the inequality symbol as an equals sign. Through linear equations, students gain a better grip on handling inequalities, grasping how linear transformations affect solutions.