Problem 19 - Precalculus Mathematics for Calculus

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$3 x+11 \leq 6 x+8$$

Step-by-Step Solution

Answer

\([1, \infty)\) is the solution.

Step 1: Set Up the Inequality

Start with the given inequality:\[ 3x + 11 \leq 6x + 8 \]

Step 2: Move Terms Involving x

To isolate the variable, move all terms involving \( x \) to one side of the inequality. Subtract \( 3x \) from both sides:\[ 3x + 11 - 3x \leq 6x + 8 - 3x \]This simplifies to:\[ 11 \leq 3x + 8 \]

Step 3: Move Constant Terms

Next, move the constant term on the right side of the inequality to the left side. Subtract \( 8 \) from both sides:\[ 11 - 8 \leq 3x + 8 - 8 \]This simplifies to:\[ 3 \leq 3x \]

Step 4: Solve for x

To isolate \( x \), divide both sides of the inequality by \( 3 \):\[ \frac{3}{3} \leq \frac{3x}{3} \]This simplifies to:\[ 1 \leq x \] Alternatively, the solution can also be written as:\[ x \geq 1 \]

Step 5: Express Solution in Interval Notation

The solution \( x \geq 1 \) can be expressed in interval notation as:\[ [1, \infty) \]

Step 6: Graph the Solution Set

Draw a number line. Shade the portion of the line starting from \( 1 \) and extending to the right towards infinity. Place a closed circle on \( 1 \) to indicate that it is included in the solution set. This graphically represents \([1, \infty)\).

Key Concepts

Interval Notation

Interval notation is a simple way to describe a set of numbers along a number line. It shows the start and end points of the range of solutions and indicates whether these points are included or not. When working with linear inequalities, interval notation is a convenient way to display solutions because it clearly defines the boundary points.
There are two types of intervals in interval notation:

• Closed Interval [a, b]: This means the interval includes both endpoints, so both "a" (the lower bound) and "b" (the upper bound) are part of the solution. Closed intervals use square brackets, like [1, 5].
• Open Interval (a, b): This set does not include the endpoints "a" and "b". Open intervals use parentheses, like (3, 7).

In solving linear inequalities, closed intervals are used when the inequality is either "less than or equal to" (≤) or "greater than or equal to" (≥). For instance, the solution to the inequality \( x \geq 1 \) is expressed as the interval [1, ∞) because it includes 1 and continues indefinitely to the right, which is represented by the infinity symbol.

Graphing Solutions

Graphing solutions of linear inequalities helps visually understand the range of values that satisfy the inequality. On a number line, it is essential to correctly represent the endpoints and determine whether they belong to the solution set.
Here's how to graph solutions:

• Draw the Number Line: Begin by drawing a horizontal line with numbers that appropriately span the range of interest.
• Identify the Endpoint: Determine if the endpoints are included in the solution by whether the inequality includes ≤ or ≥. If so, use a closed circle. For strict inequalities (< or >), use an open circle.
• Shade the Solution Set: For an inequality like \( x \geq 1 \), shade the entire region to the right of the endpoint on the number line. This shading indicates all numbers that make the inequality true.

Graphing provides a clear visual representation of solutions, making it easy to see which values of "x" satisfy the inequality. In this example, the graph for \( [1, \infty) \) begins with a filled circle at "1" and extends rightward indefinitely.

Solving Algebraic Equations

Solving algebraic equations, particularly linear inequalities, is about finding the value(s) of the variable that make the equation or inequality true. The process is similar to solving regular equations, with a few extra considerations for inequalities.
Here's a breakdown of solving an inequality like \( 3x + 11 \leq 6x + 8 \):

• Isolate the Variable: You start by simplifying the inequality, like moving terms with "x" to one side. Here, subtract 3x from both sides to focus efforts; it results in \( 11 \leq 3x + 8 \).
• Simplify Further: Once variables are mostly isolated, manage constant terms. Subtract 8 from both sides, giving \( 3 \leq 3x \).
• Solve for x: Divide both sides by the coefficient of "x" to solve for its value. After dividing by 3, \( x \geq 1 \) or seen as \( 1 \leq x \) represents our solution.

Remember, when dividing or multiplying both sides of an inequality by a negative number, you must reverse the inequality sign. Although this step wasn't necessary for our specific problem, it's a crucial rule that can change the solution set considerably.