Problem 20 - Algebra and Trigonometry

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 6-x \geq 2 x+9 $$

Step-by-Step Solution

Answer

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

Step 1: Move variable terms to one side

Begin by moving all terms involving the variable to one side of the inequality. In this case, let's move the term involving \(x\) from the right side to the left side by subtracting \(2x\) from both sides: \[6 - x - 2x \geq 9\]

Step 2: Simplify the inequality

Combine like terms on the left side of the inequality to simplify it:\[6 - 3x \geq 9\]

Step 3: Isolate the variable term

Next, we want to isolate the term containing \(x\). We can do this by subtracting \(6\) from both sides of the inequality:\[-3x \geq 3\]

Step 4: Solve for the variable

To solve for \(x\), divide both sides by \(-3\). Remember that dividing both sides of an inequality by a negative number reverses the inequality sign:\[x \leq -1\]

Step 5: Express the solution in interval notation

The inequality \(x \leq -1\) represents all numbers that are less than or equal to \(-1\). In interval notation, this is expressed as:\[(-\infty, -1]\]

Step 6: Graph the solution set

Draw a number line and shade the region to the left of \(-1\). Use a closed dot or bracket on \(-1\) to indicate that \(-1\) is included in the solution:```<----[==== -1 ```This represents all numbers less than or equal to \(-1\.\)

Key Concepts

Solving Inequalities

When it comes to solving inequalities like the one in our example, \(6-x \geq 2x+9\), the goal is to find which values of \(x\) satisfy the inequality. Think of it like balancing a scale: we want to make one side equivalent or more.To solve this problem, we need to isolate the variable on one side of the inequality.Here’s how you can approach it:

• First, ensure all terms with \(x\) are on one side. In our example, we moved \(2x\) from the right to the left by subtraction.
• Simplify any like terms. This means combining any similar terms.
• Isolate the \(x\) term by moving constants (numbers without \(x\)) to the other side.
• When dividing or multiplying by a negative number, remember to reverse the inequality sign.This is a critical step in inequalities.

By these steps, we find that \(x \leq -1\).This tells us all the numbers \(x\) could be, so long as they don't exceed \(-1\).

Interval Notation

Interval notation is a shorthand way of writing subsets of the real number line.It helps us express the solution sets of inequalities in a clean and easy-to-read format.For our inequality \(x \leq -1\), we represent it in interval notation as \((-\infty, -1]\).

• The round bracket \((\) means that \(-\infty\) is not included because infinity is not a number but an idea of unboundedness.
• The square bracket \([\) next to \(-1\) indicates that \(-1\) is included in the solutions set.

Interval notation is beneficial because it shows at a glance both the range of values being considered and whether the endpoints are included. For students learning inequalities, it's an essential tool to understand.

Graphing Inequalities

Graphing the solution to an inequality gives a visual representation, helping you to see the range of possible solutions.For this type of inequality, we use a number line to illustrate the solution.Here's how to graph \(x \leq -1\):

• Mark the point \(-1\) on the number line.
• Use a closed dot (or a bracket) to indicate \(-1\) is part of the solution.
• Shade the region to the left of \(-1\), which represents all numbers less than \(-1\).

The graph is a practical means to quickly identify all solutions to an inequality. Remember, shaded areas show where the inequality holds true, and dots or brackets indicate if the endpoint is included.