1. **Stating the problem:** We want to summarize key concepts about solving first-degree equations and inequalities with one variable. 2. **Definition:** A first-degree equation in one variable is an algebraic equality where the variable appears at most to the first power. 3. **General approach to solving:** - Group variable terms on one side and constants on the other. - Isolate the variable by performing inverse operations. 4. **Example:** Solve $$2x - 8 = -3x + 22$$ - Move variable terms to left: $$2x + 3x = 22 + 8$$ - Simplify: $$5x = 30$$ - Divide both sides by 5: $$x = \frac{30}{5} = 6$$ 5. **Verification:** Substitute $$x=6$$ back into the original equation: $$2 \times 6 - 8 = -3 \times 6 + 22$$ $$12 - 8 = -18 + 22$$ $$4 = 4$$ which is true. 6. **Infinite solutions:** Equations like $$0 \times x = 0$$ have infinitely many solutions because any $$x$$ satisfies the equation. 7. **No solution:** Equations like $$0 \times x = k$$ with $$k \neq 0$$ have no solution. 8. **Inequalities:** Similar rules apply, but when multiplying or dividing by a negative number, the inequality sign reverses. This summary captures the essential methods and cases for solving first-degree equations and inequalities.