Problem 25 - Precalculus Mathematics for Calculus

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$4-3 x \leq-(1+8 x)$$

Step-by-Step Solution

Answer

The solution is \((-\infty, -1]\).

Step 1: Distribute and Simplify

The inequality given is \(4 - 3x \leq -(1 + 8x)\). Start by distributing the negative sign on the right side: \(4 - 3x \leq -1 - 8x\). This simplifies the equation for easier manipulation.

Step 2: Add 8x to Both Sides

To eliminate \(-8x\) from the right-hand side, add \(8x\) to both sides of the inequality: \(4 - 3x + 8x \leq -1 - 8x + 8x\), which results in \(4 + 5x \leq -1\).

Step 3: Subtract 4 from Both Sides

Next, isolate the term with \(x\) by subtracting \(4\) from both sides: \(4 + 5x - 4 \leq -1 - 4\), simplifying to \(5x \leq -5\).

Step 4: Divide by 5

To solve for \(x\), divide both sides by 5: \(\frac{5x}{5} \leq \frac{-5}{5}\), resulting in \(x \leq -1\). This is the condition for \(x\).

Step 5: Express Solution in Interval Notation

The solution \(x \leq -1\) can be expressed in interval notation as \((-\infty, -1]\).

Step 6: Graph the Solution Set

On a number line, represent the solution by drawing a line from \(-\infty\) to \(-1\) and shading the line. Place a closed dot on \(-1\) to include this value in the solution set.

Key Concepts

Interval Notation

Interval notation is a concise way to represent a range of numbers, especially when dealing with inequalities. It highlights the set of all possible solutions in one simple expression. - **Brackets and Parentheses**: Brackets \[ ..., ... \] are used to include endpoints in an interval, while parentheses \( ..., ... \) exclude them. For example, \[0, 5\] means all numbers from 0 to 5, including 0 and 5. However, \(0, 5\) means numbers between 0 and 5, excluding the endpoints.- **Infinites and Unbounded Intervals**: When an interval extends infinitely in one direction, use \( \infty \) or \( -\infty \). Always use parentheses with infinity as it is not a number that can be reached or included. For instance, \((-\infty, -1]\) includes all numbers less than or equal to \(-1\), but stretches to infinity.- **Reading the Solution**: In our example, solving the inequality gives \( x \leq -1 \), translating to the interval \((-\infty, -1]\). This interval contains every number up to and including \(-1\), covering all values that \(x\) can take.

Number Line Graph

A number line graph is a visual representation that helps to easily see the range of solutions to an inequality. It provides an immediate comprehension of where all possible solutions lie.- **Basic Layout**: A number line is a horizontal line marked with numbers in order of increasing value. Each point on this line corresponds to a number.- **Shading and Symbols**: For an inequality like \(x \leq -1\), draw a line extending from \(-\infty\) to \(-1\). Shade this line to show all included numbers. At \(-1\), place a closed dot to signify that \(-1\) is part of the solution set.- **Importance and Usage**: Using the number line makes it straightforward to interpret and communicate the extent of solutions. For students, it's a helpful tool to check their solution visually after solving the inequality.

Solving Inequalities

Solving inequalities is crucial for finding possible values that satisfy conditions set by an inequality. Here's a step-by-step breakdown:- **Simplification**: Start by simplifying the inequality. Distribute any factors, and combine like terms on both sides. In our inequality \(4 - 3x \leq -(1 + 8x)\), distribute the negative to simplify to \(4 - 3x \leq -1 - 8x\).- **Balancing both sides**: Add or subtract terms on both sides to isolate terms involving the variable. Add \(8x\) to both sides to cancel it out on the right, simplifying to \(4 + 5x \leq -1\).- **Isolating the Variable**: Get the variable term by itself. Subtract \(4\) from both sides to continue isolating \(x\), which gives \(5x \leq -5\).- **Solving for the Variable**: Finally, divide to solve for the variable. Divide both sides by 5 to find \(x \leq -1\).- **Applying Solution**: With \(x \leq -1\), express it in interval notation as \((-\infty, -1]\). Graph it to visually confirm the solution.Understanding these steps ensures that you can solve any linear inequality with confidence.