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When solving inequalities, the goal is to find all values of the variable that make the inequality true. Let's take the example of the inequality given: \(8x + 9 leq -15\).
Follow these steps:

• Isolate the variable term:

Subtract 9 from both sides: \(8x + 9 - 9 leq -15 - 9\), which simplifies to \(8x leq -24\).

• Solve for the variable:

Divide both sides by 8: \(\frac{8x}{8} leq \frac{-24}{8}\), which simplifies to \(x leq -3\).
In short, the solution for \(x\) without further unknowns or fractions is called an inequality solution. You are finding the set of all numbers that, when substituted into the inequality, make it true.

Interval notation is a shorthand used to represent sets of numbers on the number line.
It tells you which parts of the number line belong to the solution set of the inequality.
For the inequality solution \(x leq -3\) we use interval notation to represent the range of values:

• An interval that is unbounded to the left (going towards negative infinity) and
• A point where it is inclusive or exclusive on the right end of the interval.

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Here, -3 is included, so the interval notation is \(( -\infty, -3 ]\). When -3 is included, we use a square bracket \([\) on the side of -3.
For points which aren't included, like when you use \(<\) or \(>\), you would use a parenthesis \(()\). Remember that

• \( (-\infty, -3 )\) means \( x < -3 \)
• \( (-\infty, -3 ]\) means \( x \leq -3 \)

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You can easily communicate the solution of an inequality using this notation.

To visually represent the solution of an inequality, we can graph it on a number line.

• Draw a number line:

It should extend to negative and positive infinity. Mark the critical point mentioned in your solution, in this case, -3.

• Use the proper symbol for the critical point:

Since -3 is included (because \(x \leq -3\)), put a closed dot (鈼�) at -3.

• Shade the correct side:

To show all values less than or equal to -3 (\(x \leq -3\)), shade the region to the left of -3, extending towards negative infinity. This visually confirms the values of \(x\) that satisfy the inequality.In summary, graphing inequalities helps you easily see which ranges of values make the inequality true. Graphical representation complements interval notation and provides an intuitive understanding of the solution set.