Problem 60 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -22<5 x+3 \leq 3 $$

Step-by-Step Solution

Answer

The solution set is \((-5, 0]\).

Step 1: Break Down the Compound Inequality

The compound inequality is given as \[-22 < 5x + 3 \leq 3\].This represents two separate inequalities:1. \(-22 < 5x + 3\)2. \(5x + 3 \leq 3\).

Step 2: Solve the First Inequality

Start by solving the first inequality:\[-22 < 5x + 3\].Subtract 3 from both sides:\[-22 - 3 < 5x\],which simplifies to\[-25 < 5x\].Now divide both sides by 5:\[-5 < x\].

Step 3: Solve the Second Inequality

Now solve the second inequality:\[5x + 3 \leq 3\].Subtract 3 from both sides:\[5x \leq 0\].Now divide both sides by 5:\[x \leq 0\].

Step 4: Combine the Inequalities

Combine the solutions from both inequalities:\(-5 < x\) and \(x \leq 0\).This results in the combined inequality:\[-5 < x \leq 0\].

Step 5: Write the Solution in Interval Notation

Express the solution \(-5 < x \leq 0\) in interval notation:\((-5, 0]\).

Step 6: Graph the Solution

To graph the solution set \((-5, 0]\), draw a number line. Place an open circle (meaning 'not including') at -5 and a closed circle (meaning 'including') at 0. Shade the region between -5 and 0, moving towards the right.

Key Concepts

Graphing Inequalities

Graphing inequalities is a visual way to understand the range of values an expression can take. Think of the number line as your canvas. Each point on the number line represents a number your variable "x" could be.

To begin, identify the critical points from your solution space. In this context, these are the boundaries where the inequality changes from true to false, or vice versa. Using the solution \((-5 < x \leq 0)\), we have two critical points: -5 and 0.

- At -5, use an open circle, indicating that this value is not included in the solution set.- At 0, a closed circle is drawn to indicate that 0 is included in the solution.

Once you have plotted these points, shade the region of the graph between -5 and 0. This shaded area represents all possible values of "x" that satisfy the inequality. Graphing serves as a powerful tool to visualize constraints placed upon the variable.

Compound Inequalities

Compound inequalities involve two or more separate inequalities being combined. They allow us to define a more complex solution range than with a single inequality. In this context, we started with the compound inequality \(-22 < 5x + 3 \leq 3\). This was divided into two simpler parts:
- \(-22 < 5x + 3\) - \(5x + 3 \leq 3\)

To solve these, break them down one by one. Simplifying them involves the same process as solving regular linear equations:

• Subtract any constant terms from both sides.
• Divide or multiply by the coefficient of "x" to isolate the variable.

The individual solutions, \(-5 < x\) and \(x \leq 0\), are then combined to form the compound solution \(-5 < x \leq 0\).

This intersection of solutions means "x" must satisfy both conditions simultaneously, offering a precise region on the number line.

Interval Notation

Interval notation is a concise way of expressing a solution set. Instead of writing out words like "greater than" and "less than," we use symbols and numbers to show which parts of the number line are included. For the combined inequality solution \(-5 < x \leq 0\), the interval notation is written as \((-5, 0]\).

Here's how to understand the symbols:

• Parentheses \(()\) mean that the boundary number is not included in the set. For example, in \((-5, 0]\), the -5 is not included.
• Brackets \([]\) mean that the boundary number is included. In \((-5, 0]\), the 0 is included because of the bracket.

Remember, interval notation helps streamline writing and understanding solution sets, especially when dealing with ranges of numbers on the real number line.