Problem 25 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 2 \leq x+5<4 $$

Step-by-Step Solution

Answer

The solution is \([-3, -1)\).

Step 1: Understanding the inequality

The given inequality is a compound inequality: \( 2 \leq x + 5 < 4 \). This means we need to solve it in two parts: \( 2 \leq x + 5 \) and \( x + 5 < 4 \). Both conditions need to be true at the same time.

Step 2: Solve the first inequality

Let's solve \( 2 \leq x + 5 \) by isolating \( x \). Subtract 5 from both sides: \( 2 - 5 \leq x + 5 - 5 \). This simplifies to \( -3 \leq x \).

Step 3: Solve the second inequality

Now, solve \( x + 5 < 4 \) by subtracting 5 from both sides: \( x + 5 - 5 < 4 - 5 \). This simplifies to \( x < -1 \).

Step 4: Combine the solutions

The solution to the compound inequality is formed by combining the results from Step 2 and Step 3. Hence, \(-3 \leq x < -1\).

Step 5: Represent the solution in interval notation

The interval notation for the solution \(-3 \leq x < -1\) is \([-3, -1)\). This means \(x\) includes \(-3\) but not \(-1\).

Step 6: Graph the solution set

To graph \([-3, -1)\), draw a number line and make a closed dot on \(-3\) and an open dot on \(-1\). Shade the region between these two dots to indicate all numbers \(x\) that satisfy the inequality \(-3 \leq x < -1\).

Key Concepts

Interval Notation

Interval notation is a way to express a set of numbers between two endpoints. It is helpful when describing the solution set of inequalities, especially linear ones. There are different symbols used in interval notation to indicate whether an endpoint is included or excluded from the set:

• A square bracket [ ] means the endpoint is included in the interval. It shows that the number at the endpoint is part of the solution.
• A parenthesis ( ) indicates that the endpoint is not included. This means that the number is just outside the interval's boundary.

For example, the interval notation \([-3, -1)\) represents all numbers starting from \-3\ and going up to but not including \-1\. This particular example tells us that \-3\ is included, while \-1\ is not. Interval notation is a concise and straightforward way to capture solutions from inequalities, allowing students to visualize possible values and incorporate end points easily.

Compound Inequality

A compound inequality involves two or more simple inequalities joined by the word "and" or "or". The solution to a compound inequality "and" type must satisfy all parts of the inequality simultaneously.
For instance, the expression \( 2 \leq x+5 < 4 \) is a compound inequality. It can be split into two conditions: \( 2 \leq x+5 \) and \( x+5 < 4 \).

To find the solution, solve each inequality individually:

• First inequality: \( 2 \leq x+5 \). Subtract 5 from both sides to get \( -3 \leq x \).
• Second inequality: \( x+5 < 4 \). Subtract 5 from both sides to get \( x < -1 \).

Combining these solutions gives the range where both conditions are true, resulting in \( -3 \leq x < -1 \).
Compound inequalities provide structure to understand multiple conditions together and are especially useful in solving and interpreting mathematical problems involving ranges.

Number Line Graphing

Graphing solutions on a number line is a practical way to visually express the set of solutions for an inequality. It helps highlight which numbers are included in the solution and how the solution is distributed along the real number line.
To graph \([-3, -1)\):

• Draw a straight horizontal line to represent the number line itself.
• Identify and mark the endpoints. Place a closed dot at -3, meaning that -3 is included in the solution. This is because the inequality shows \( -3 \leq x \).
• Place an open dot at -1 to show that -1 is excluded from the solution, complying with the inequality \( x < -1 \).
• Shade the region on the number line between these two dots to indicate all numbers \( x \) that satisfy the inequality \( -3 \leq x < -1 \).

Number line graphing is an excellent tool for academics to visually translate mathematical solutions, making them more approachable and interpretable.