Solve each inequality. Give the solution set in interval notation. 2-4x+5(x-1)<-6(x-2)

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Start by expanding the expressions on both sides of the inequality. Distribute the 5 on the left side and the -6 on the right side: $2 - 4x + 5(x - 1) < -6(x - 2)$ becomes $2 - 4x + 5x - 5 < -6x + 12$.

Combine like terms on the left side: $2 - 4x + 5x - 5$ simplifies to $(2 - 5) + (-4x + 5x)$, which is $-3 + x$.

Rewrite the inequality with the simplified left side: $-3 + x < -6x + 12$.

Next, get all variable terms on one side and constants on the other. Add \$6x$ to both sides and add \(3$ to both sides to isolate \)x$: $x + 6x < 12 + 3$ which simplifies to $7x < 15$.

Finally, solve for $x$ by dividing both sides by 7: $x < \frac{15}{7}$. Express the solution set in interval notation as $(-\infty, \frac{15}{7})$.

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Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side to determine the range of values that satisfy the inequality. Similar to equations, operations like addition, subtraction, multiplication, and division are used, but multiplying or dividing by a negative number reverses the inequality sign.

Distributive Property

The distributive property allows you to multiply a single term across terms inside parentheses, such as a(b + c) = ab + ac. This is essential for simplifying expressions before solving inequalities, ensuring all terms are combined correctly.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets to indicate open or closed intervals. For example, (a, b) denotes all numbers between a and b, excluding endpoints, while [a, b] includes them.