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Solving inequalities involves a few key steps that help you find the valid range of values for an unknown variable. Let's break down these steps using the example exercise: $$-1 - 4x \geq 7$$.
First, isolate the variable term. To do this, you need to move the constant on the same side as the variable so that the equation becomes simpler. Add 1 to both sides:
\begin{aligned}-1 - 4x + 1 &\geq 7 + 1 \ -4x &\geq 8, \text{{Second, solve for the variable by isolating it. This often involves division or multiplication. Note that when multiplying or dividing by a negative number, the inequality sign flips. In this example, divide both sides by -4:}} \ \frac{-4x}{-4} &\leq \frac{8}{-4} \ {x} &\leq -2, \text{{Finally, verify and simplify the expression where required. What you have now is the solution}}{-4x \leq 8}.Isolating the variable and flipping the inequality sign upon division or multiplication by a negative are crucial points to remember when solving inequalities. Always double-check your steps to ensure the solution holds true.

Interval notation is a way to express the set of all numbers that satisfy an inequality. It's a concise way of writing continuous ranges of numbers.

For our inequality, \x \ \leq -2, the solution set includes all numbers that are less than or equal to -2. In interval notation, we write this as: \((-\infty, -2]\).

Here:

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