91Ó°ÊÓ

Algebraic operations are key steps that help in solving mathematical problems, especially inequalities. In this example, we deal with an inequality: \(1 \leq x+3 \leq 5\).
To solve this, we need to isolate the variable \(x\). This is done by performing the same operation on all parts of the inequality.
Here, the operation is to subtract 3 from each part. Always ensure you perform this uniformly to maintain the balance of the inequality:

• Subtract 3 from the left side: \(1 - 3 = -2\).
• Subtract 3 from the middle part: \(x+3 - 3 = x\).
• Subtract 3 from the right side: \(5 - 3 = 2\).

After performing these operations, we end up with a new simplified inequality: \(-2 \leq x \leq 2\).
Algebraic operations like addition, subtraction, multiplication, and division help us manipulate these inequalities to find the range where the variable lies.

After simplifying the inequality, you need to represent it visually on a number line. This helps in understanding the range of values that the variable can take. For the inequality \(-2 \leq x \leq 2\):

• Draw a straight horizontal line.
• Mark points on the number line corresponding to integers including -2 and 2.
• Shade the region between -2 and 2 to indicate the range of values where the inequality holds true.
• Use solid dots at -2 and 2. This indicates that these points are included in the inequality.

This visual representation makes it easier to comprehend the solutions of the inequality and quickly see the valid range of the variable.
It is also helpful for comparing inequalities and understanding overlapping ranges.

Understanding the range of values in inequalities is crucial as it tells us all possible values the variable can take while satisfying the condition. For the inequality \(-2 \leq x \leq 2\):

• The variable \(x\) can take any value within the range -2 and 2.
• This includes all integers, fractions, and decimals between -2 and 2.
• Both -2 and 2 are included since the inequality is 'less than or equal to' (\(\leq\)).

Expressing the range clearly, as in this example, helps to understand the solution set of the inequality. When dealing with inequalities, always pay attention to whether the endpoints are included (using \(\leq\) or \(\geq\)) or excluded (using \( < \) or \( > \)).