Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 39

Chapter 2, Problem 39

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. 1 - (x + 3) ≥ 4 - 2x

Verified step by step guidance

Start by rewriting the inequality clearly: $1 - (x + 3) \geq 4 - 2x$.

Distribute the negative sign across the parentheses on the left side: $1 - x - 3 \geq 4 - 2x$.

Combine like terms on the left side: $(1 - 3) - x \geq 4 - 2x$ which simplifies to $-2 - x \geq 4 - 2x$.

Add \$2x$ to both sides to get all $x$ terms on one side: $-2 - x + 2x \geq 4 - 2x + 2x$, simplifying to $-2 + x \geq 4$.

Add 2 to both sides to isolate the $x$ term: $-2 + x + 2 \geq 4 + 2$, which simplifies to $x \geq 6$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side to find the range of values that satisfy the inequality. Similar to equations, you perform operations like addition, subtraction, multiplication, or division, but must reverse the inequality sign when multiplying or dividing by a negative number.

Interval Notation

Interval notation is a concise way to represent sets of numbers that satisfy inequalities. It uses parentheses () for values not included and brackets [] for values included, indicating the start and end points of the solution set on the number line.

Graphing Solution Sets on a Number Line

Graphing solution sets involves marking the values that satisfy the inequality on a number line. Use open circles for values not included and closed circles for included values, shading the region that represents all possible solutions.

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