Problem 39 - College Algebra

Question

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$1-(x+3) \geq 4-2 x$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 6\), graphed on a number line with a closed dot at \(6\) and an arrow pointing to the right. The interval notation for this solution is \([6, \infty)\).

Step 1: Simplify the Inequality

Distribute the negative sign in the left side of the inequality: \(1 - x - 3 \geq 4 - 2x\), which simplifies to \(-x - 2 \geq 4 - 2x\)

Step 2: Combine Like Terms

Add \(2x\) to both sides of the inequality to collect the x terms on one side and the constants on the other side: \(x - 2 \geq 4\)

Step 3: Final Simplification

To isolate \(x\), add \(2\) to both sides of the inequality: \(x \geq 6\)

Step 4: Graph the Solution and Express in Interval Notation

Plot the solution on a number line with a closed dot at \(6\) (because \(x\) can be \(6\)) and an arrow pointing to the right (because \(x\) can be any number greater than \(6\)). The interval notation for this solution is \([6, \infty)\).

Key Concepts

Linear Inequality Graphing

When it comes to visualizing the solutions of linear inequalities, graphing on a number line offers a clear picture of all the possible values that satisfy the inequality.

For example, consider the inequality \(x \geq 6\). To graph this, we place a closed dot on the number line at \(6\) to indicate that \(6\) is included in the solution set. Using a closed dot is crucial as it signifies that the endpoint is part of the solution. Then, we draw an arrow extending to the right to show that the solution includes every number greater than \(6\) as well.

Understanding these graphical representations is essential as they provide a visual method to check your answers and can help in understanding the relationships between numbers in different types of inequalities.

Interval Notation

Interval notation is a concise way of writing sets of numbers that are solutions to inequalities. In interval notation, we use brackets \([]\) and parentheses \(()\) to describe intervals. The bracket \([\) indicates that the number is included in the set, similar to a closed dot on the graph, while the parenthesis \(()\) suggests the number is not included, which is depicted by an open dot.

For the inequality \(x \geq 6\), the solution in interval notation is \([6, \infty)\) where the square bracket at \(6\) implies that \(6\) is part of the solution, and the parenthesis at \(\infty\) indicates that the solution continues indefinitely to the right. Interval notation is an efficient tool, making it easier for us to write and read large sets of numbers in a standardized form.

Inequality Simplification

Simplifying inequalities is a step-by-step process similar to simplifying equations, but it's crucial to pay attention to the direction of the inequality.

To simplify an inequality like \(1-(x+3) \geq 4-2x\), you start by distributing any factors and combining like terms. In this case, distributing the negative across \(x+3\) transforms the inequality into \(-x - 2 \geq 4 - 2x\). Afterward, it's a matter of moving all terms involving \(x\) to one side and constants to the other, yielding \(x \geq 6\) as in our example.

Always remember that if you multiply or divide the inequality by a negative number, the direction of the inequality will change. However, in our example, we avoided this by adding \(2x\) and \(2\) to both sides, maintaining the correct inequality direction throughout the simplification process.