In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 2x - 11 < - 3(x + 2)

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Start by distributing the -3 on the right side of the inequality: $2x - 11 < -3(x + 2)$ becomes $2x - 11 < -3x - 6$.

Next, add \$3x$ to both sides to get all the $x$ terms on one side: $2x + 3x - 11 < -3x + 3x - 6$ which simplifies to $5x - 11 < -6$.

Then, add 11 to both sides to isolate the term with $x$: $5x - 11 + 11 < -6 + 11$ which simplifies to $5x < 5$.

Now, divide both sides by 5 to solve for $x$: $\frac{5x}{5} < \frac{5}{5}$ which simplifies to $x < 1$.

Express the solution in interval notation as $(-\infty, 1)$ and graph this on a number line by shading all values less than 1 with an open circle at 1.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Solving Linear Inequalities

A linear inequality involves an inequality symbol (<, >, ≤, ≥) with a linear expression. To solve it, isolate the variable by performing algebraic operations similar to equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

Interval Notation

Interval notation is a way to represent solution sets of inequalities using parentheses and brackets. Parentheses indicate that an endpoint is not included, while brackets mean it is included. For example, (a, b) means all numbers between a and b, excluding a and b.

Graphing Solution Sets on a Number Line

Graphing solution sets involves marking the range of values that satisfy the inequality on a number line. Use open circles for values not included and closed circles for included endpoints. Shade the region representing all solutions to visually communicate the solution set.