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Chapter 2: Problem 57

Solve each inequality and express the solution set using interval notation. \(-3(3 x+2)-2(4 x+1) \geq 0\)

Short Answer

Expert verified
The solution set is \((-\infty, -\frac{8}{17}]\).

Step by step solution

01

Simplify the Left Side

First, distribute the constants inside each set of parentheses in the inequality \(-3(3x+2) - 2(4x+1) \geq 0\). This results in: \(-9x - 6 - 8x - 2 \geq 0\).
02

Combine Like Terms

Combine the like terms on the left side of the inequality: \(-9x - 8x - 6 - 2 \geq 0\) simplifies to \(-17x - 8 \geq 0\).
03

Isolate the Variable

Add 8 to both sides to isolate the term containing \(x\):\(-17x \geq 8\).
04

Solve for x

Divide both sides by -17, remembering to flip the inequality sign,resulting in: \(x \leq -\frac{8}{17}\).
05

Express Solution in Interval Notation

The solution \(x \leq -\frac{8}{17}\) translates to the interval \((-\infty, -\frac{8}{17}]\) in interval notation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Interval notation is a way of writing subsets of the real number line. It allows us to describe a range of values succinctly and clearly. The interval notion consists of opening and closing brackets, numbers, and sometimes symbols like infinity
  • Brackets: "(" and ")" specify that an endpoint is not included, called open intervals.
  • Square brackets: "[" and "]" specify that an endpoint is included, which are closed intervals.
For example, the interval \((-\infty , -\frac{8}{17} ]\) means all numbers less than or equal to \-\frac{8}{17}\. The "-\infty" symbolizes no lower limit and a parenthesis indicates it's not part of the interval.
This method is a quick and effective way to convey solutions to inequalities, especially when the solutions comprise a continuous set of numbers.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations that represent a value. In the inequality \[-3(3x+2) - 2(4x+1) \geq 0\], we have an algebraic expression on the left side.
These expressions can involve one or more terms:
  • Terms are separated by plus or minus signs.
  • Each term consists of numbers and variables multiplied together, such as \(-9x\) or \(-6\).
Simplifying algebraic expressions involves distributing, combining like terms, and performing arithmetic operations. This makes them easier to work with or solve.
Understanding expressions is crucial because they form the building blocks for equations and inequalities.
Inequality Step-by-Step Solution
Solving inequalities step-by-step ensures every part of the process is clear and understandable. Here's a breakdown of our problem's solution:
  • Step 1: Distribute numbers across each set of parentheses: \(-3(3x+2)\) becomes \(-9x-6\) and \(-2(4x+1)\) becomes \(-8x-2\).
  • Step 2: Combine the like terms: In \(-9x-6-8x-2\), the terms \(-9x-8x\) become \(-17x\). The numbers \(-6-2\) combine to \(-8\).
  • Step 3: Isolate the variable by adding 8: Rewriting \(-17x-8 \geq 0\), we add 8 to both sides to get \(-17x \geq 8\).
  • Step 4: Solve for \(x\) by dividing: Divide both sides by \(-17\). Swap the inequality sign to obtain \(x \leq -\frac{8}{17}\).
Each step shows how simplifying, combining, and calculating communicate the path to solving inequalities effectively.
This structure helps break down complex inequalities into manageable bits, providing a map from start to solution.

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Most popular questions from this chapter

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ 0<2 x+5<12 $$

Explain how you would solve the inequality \(|2 x+5|>-3\).

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -3<2-x<3 $$

Solve each inequality. $$ |1-2 x|<2 $$

Solve each equation. $$ |-2 x-3|=|x+1| $$
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