Problem 14 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. Write each answer using solution set notation. $$ x+4 \leq 1 $$

Step-by-Step Solution

Answer

The solution set is \( \{ x \mid x \leq -3 \} \).

Step 1: Subtract 4 from both sides

To isolate the variable \( x \) on the left side of the inequality, subtract 4 from both sides. This gives us \( x + 4 - 4 \leq 1 - 4 \). Simplifying both sides results in \( x \leq -3 \).

Step 2: Graph the inequality

On a number line, locate the point \(-3\). Since this is a 'less than or equal to' inequality, place a closed circle on \(-3\) to include it. Shade all points to the left of \(-3\) to indicate that \( x \) can be any number less than or equal to \(-3\).

Step 3: Write the solution in set notation

The solution set includes all values of \( x \) that satisfy the inequality. In set notation, this is written as \( \{ x \mid x \leq -3 \} \). This reads as 'the set of all \( x \) such that \( x \) is less than or equal to \(-3\)'.

Key Concepts

Understanding the Solution Set

When solving inequalities, the solution set is a group of values that satisfy the given condition. For the inequality \( x + 4 \leq 1 \), our goal is to find all possible values of \( x \) that make the inequality true. By subtracting 4 from both sides, we simplify the inequality to \( x \leq -3 \). This tells us that any number that is less than or equal to \(-3\) is part of the solution set.
A solution set can be thought of as the answer to the inequality. It's like saying, "Here are all the values that will work!" In this case, it's any number that is \(-3\) or lower. When tackling inequalities, always aim to isolate the variable to clearly see what values meet the condition.

Graphing Inequalities on a Number Line

Graphing inequalities helps visualize the solution set. For the inequality \( x \leq -3 \), we begin by locating \(-3\) on the number line.

• Since we have a 'less than or equal to' situation, we use a closed circle or dot on \(-3\). This shows that \(-3\) is included in the solution set.
• Next, we shade or draw an arrow to the left of \(-3\), indicating all values less than \(-3\) are part of the solution. The shading doesn't stop; it goes infinitely in the negative direction, representing all numbers less than \(-3\).

This graph provides a quick visual of which numbers are solutions, making it easier to understand the extent of the solution set. Every point on the shaded side of the number line is a number that satisfies \( x \leq -3 \).

Using Set Notation

Set notation is a precise way to express a solution set. For our inequality, the solution is written in set notation as \( \{ x \mid x \leq -3 \} \). This might look fancy, but it's saying something quite simple.

• The curly braces \( \{ \} \) indicate that we are describing a set, or a collection, of numbers.
• The vertical bar \( \mid \) means "such that." So, the whole expression reads: "The set of all \( x \) such that \( x \) is less than or equal to \(-3\)."

Set notation is standardized, making it a universal language among mathematicians. It clearly communicates the full range of numbers that satisfy the inequality, without needing a long explanation. Whether you use graphs or set notation, the goal is the same: to show all the solutions to the inequality.