We are solving the inequality:

$x - 3 \leq 6x + 7$

Step 1: Simplify the inequality

Move all terms involving $x$ to one side and constants to the other:

$x - 6x \leq 7 + 3$

$-5x \leq 10$

Step 2: Solve for $x$

Divide both sides by $-5$. Remember, when dividing or multiplying by a negative number, the inequality sign flips:

$x \geq -2$

Step 3: Write the solution set in interval notation

The solution is all values of $x$ greater than or equal to $-2$. In interval notation, this is:

$[-2, \infty)$

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Let me know if you have further questions or want a detailed explanation!

Related Questions:

1. What is the solution to $2x - 5 \leq 7x + 3$ in interval notation?
2. How does dividing by a negative value affect inequalities?
3. Solve and graph $x + 4 > 2x - 6$.
4. Can inequalities have no solution? Provide an example.
5. Write $x \geq -2$ as a graph on a number line.

Tip:

When solving inequalities, always double-check the direction of the inequality if you divide or multiply by a negative number.