Problem 63 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$-3(x+11) \leq 6$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq -13\). This is represented graphically by a closed circle at -13 with an arrow pointing to the right on a number line.

Step 1: Distribute -3 Across the Left Side of the Inequality

Multiply each term inside the parentheses by -3. This results in: \(-3x -33 \leq 6\)

Step 2: Isolate the Variable

Add 33 to both sides of the inequality to isolate the variable x. This results in \(-3x \leq 39\). Then, divide by -3 to get \(x \geq -13\).

Step 3: Graph the Solution on a Number Line

Draw a number line and plot the point -13. The inequality is greater than or equal to, which means the solution includes -13 and all numbers greater than -13. This is represented by a closed (solid) circle at -13 and an arrow pointing to the right.

Key Concepts

Inequality Graphing

Graphing inequalities allows us to visually represent the set of all possible solutions to an inequality on a number line or coordinate plane. When dealing with a simple inequality like the one given \( -3(x+11) \leq 6 \), translating this algebraically solved inequality into a graphical form helps students to easily understand and interpret the solution.

The process of graphing an inequality involves identifying whether the variable's value is greater than, less than, including (\geq), or excluding (>). Then, represent this on a number line using open or closed circles and arrows. In our example, after solving the inequality step-by-step, one would graph \( x \geq -13 \) by placing a closed circle at \(-13\) to include this value, and drawing an arrow to the right to show that the solution includes all numbers greater than or equal to \(-13\).

Distributive Property

The distributive property is a critical algebraic concept that allows us to multiply a single term by each term inside a set of parentheses. When facing an inequality such as \( -3(x+11) \leq 6 \), applying the distributive property correctly is key to simplifying the inequality.

To apply this property in our exercise, we multiply -3 by each term inside the parentheses:

• \( -3 \times x = -3x \)
• \( -3 \times 11 = -33 \)

So, our inequality becomes \( -3x - 33 \leq 6 \). This step sets the stage for further manipulation to isolate the variable and find the solution.

Isolating Variables

Isolating the variable in an inequality involves moving all terms with the variable to one side and constants to the other, to find the solution set for the variable. In our inequality \( -3(x+11) \leq 6 \), after using the distributive property, we need to isolate \(x\) to solve.

This is achieved by performing the same mathematical operation on both sides of the inequality, maintaining its balance. We first add 33 to both sides, resulting in \( -3x \leq 39 \). Then, because multiplication or division by a negative number reverses the inequality, dividing both sides by -3 flips the inequality sign yielding the solution \( x \geq -13 \). It's important for students to remember this rule when working with inequalities.

Number Line Representation

Representing inequalities on a number line is a tangible way to understand the range of solutions. After solving an inequality, such as \( x \geq -13 \), depicting it on a number line makes the set of possible values for \(x\) explicit.

A number line graph includes:

• Using a closed circle to indicate that a number is part of the solution set (\geq or \leq).
• Using an open circle when the number is not included (> or <).
• Drawing an arrow in the direction of the set of numbers that are also part of the solution.

In our worked example, the number line would show a closed circle at -13 and an arrow pointing to the right, indicating all numbers greater than or equal to -13 are included in the solution.