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When solving inequalities like \(-11 - 6x \geq 33\), it's not enough to just find a solution; we must also graph it to clearly visualize the solution set. Graphing inequalities on a number line helps us easily see which numbers satisfy the inequality.
First, determine the solution to the inequality. In this example, we found that \(x \leq -\frac{22}{3}\). On the graph, this involves:

• Locating \(-\frac{22}{3}\) on the number line.
• Using a filled-in circle at \(-\frac{22}{3}\) to show that this number is included in the solution.
• Drawing a line extending to the left from \(-\frac{22}{3}\), representing all numbers less than or equal to \(-\frac{22}{3}\).

Remember, a filled-in circle signifies "less than or equal to" or "greater than or equal to," while an open circle indicates strict inequalities like "less than" or "greater than." This visual representation is a powerful tool in understanding inequalities, making it easier to interpret and communicate the solutions.

Solving linear inequalities is much like solving linear equations, but with an extra step to remember: reversing the inequality sign when multiplying or dividing by negative numbers.
A linear inequality relates a linear expression, composed of variables and constants, to another quantity using inequality signs: \(<, \leq, >, \geq\). In our example \(-11 - 6x \geq 33\), we first isolate the variable term by adding 11 to both sides, resulting in \(-6x \geq 44\).
Since the coefficient of \(x\) is negative, we divide both sides by \(-6\). This reverses the inequality sign, turning it into \(x \leq -\frac{22}{3}\).
Solving linear inequalities follows these general steps:

• Isolate the variable on one side of the inequality.
• Perform the same operation on both sides.
  • Pay attention to reversing the inequality sign if you multiply or divide by a negative number.
• Simplify to find the solution.

Being mindful of these steps will ensure you accurately solve and interpret the meaning of linear inequalities.

The real number line is a useful tool for visualizing numbers and solutions in mathematics, particularly when dealing with inequalities.
A number line is a horizontal representation where each point corresponds to a real number, extending infinitely in both directions.
When graphing inequalities, the real number line allows us to:

• Indicate specific values using points or circles.
• Show a range of solutions by shading a section of the line.
• Easily compare values relative to each other.

For our inequality solution \(x \leq -\frac{22}{3}\), using the real number line, we:

• Placed a filled-in circle on \(-\frac{22}{3}\).
• Shaded the line leading to the left, illustrating all numbers less than or equal to \(-\frac{22}{3}\).

The real number line not only helps identify solutions but also aids in understanding relationships between numbers, vital in communicating mathematical ideas clearly and effectively.