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Interval notation is a concise method to represent a range of values, commonly used when solving inequalities. In solving the inequality \(-7x + 3 \leq 4-x\), we determined that \(x \geq -\frac{1}{6}\) is the solution. Interval notation captures this solution set neatly.

• The square bracket \([-\frac{1}{6},\) indicates that \(-\frac{1}{6}\) is included in the solution set, known as the "closed interval".
• The parenthesis \(\infty)\) signifies that the range extends indefinitely towards positive infinity and infinity is always approached, never reached, thus an "open interval".

Using interval notation saves space and provides clarity. Instead of writing out a lengthy description of all possible values, you provide boundaries of the solution set succinctly.

Graphing solution sets visually demonstrates which values satisfy the inequality. Here, in the inequality \(-7x + 3 \leq 4 - x\), the solution \([-\frac{1}{6}, \infty)\) was obtained.

• Begin by drawing a horizontal line to represent the number line.
• Place a filled circle directly at \(-\frac{1}{6}\) to denote that \(-\frac{1}{6}\) is part of the solution set.
• Shade the region of the number line extending to the right of \(-\frac{1}{6}\), indicating all greater numbers are included in the solution.

This graphical representation helps in visualizing the range of numbers that satisfy our inequality. It is particularly useful for checking or communicating solutions quickly. The filled circle at \(-\frac{1}{6}\) visually reaffirms its inclusion in the interval.

Solving linear inequalities follows specific steps that resemble solving linear equations, with a critical addition regarding negative numbers. Here's how we solved the inequality:

1. **Move Variable Terms:** - Combine like terms to keep the variable on one side of the inequality. We added \(x\) to both sides to consolidate \(x\) terms on one side.
2. **Isolate the Constant:** - Remove constant terms from the variable side by performing inverse operations. Subtracting \(3\) helped us isolate the term with \(x\).
3. **Solve for the Variable:** - Divide by the coefficient of \(x\), ensuring to reverse the inequality sign if dividing by a negative number. This step resulted in \(x \geq -\frac{1}{6}\).

Following these steps for each problem helps in consistently finding correct solutions. The special rule about reversing the inequality sign is crucial when dealing with negative multiplication or division.