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Chapter 8: Problem 36

Graph the solution set, and write it using interval notation. $$ -(9+x)-5+4 x \geq 4 $$

Short Answer

Expert verified
x 鈮� 6; Interval Notation: [6, 鈭�)

Step by step solution

01

Simplify the equation

Start by distributing and combining like terms. The given equation is - (9 + x) - 5 + 4x Simplify to: -9 - x - 5 + 4x Combine like terms: -14 + 3x So the original inequality becomes: -14 + 3x 鈮� 4
02

Isolate the variable

Add 14 to both sides to isolate the term with the variable: -14 + 3x + 14 鈮� 4 + 14 Simplifies to: 3x 鈮� 18
03

Solve for x

Divide both sides by 3 to isolate x: 3x/3 鈮� 18/3 This simplifies to: x 鈮� 6
04

Graph the solution

Draw a number line and shade the area to the right of 6, including 6 itself (filled in circle). This shows that x can take any value equal or greater than 6.
05

Write using interval notation

The interval notation for this solution set is: [6, 鈭�)

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Key Concepts

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

simplifying algebraic expressions
Let's start by looking at how to simplify algebraic expressions. This is essential because it makes the inequality much easier to handle. Take the given problem: -(9 + x) - 5 + 4x First, distribute the negative sign across the parentheses: -9 - x - 5 + 4x Now, combine like terms. The terms -9 and -5 are constants that can be added together. The terms -x and 4x are like terms involving x: -14 + 3x After simplification, the expression becomes: -14 + 3x.
isolating variables
Isolating the variable is the next crucial step. Our goal is to get x by itself on one side of the inequality. From our simplified inequality -14 + 3x 鈮� 4, add 14 to both sides to move the constant term. This results in: 3x 鈮� 18 Now, divide both sides by 3 to isolate x: x 鈮� 6. By doing this, we have effectively isolated the variable, which means we can now easily understand the range of values that x can take.
interval notation
Interval notation is a way to represent a set of numbers along an axis. For x 鈮� 6, we use the symbol [6, 鈭�). The square bracket [6, indicates that 6 is included in the solution set (closed interval), and the round bracket at 鈭�) indicates that infinity is not a number we reach, but a concept that the numbers go on forever (open interval). Interval notation simplifies the representation of solution sets, making them easier to interpret and work with.
graphing inequalities
Graphing inequalities on a number line helps visually demonstrate the range of values that satisfy the inequality. For x 鈮� 6, draw a number line and place a filled-in circle at 6 to show that 6 is included. Then shade the entire area to the right of 6 to represent all numbers greater than or equal to 6. This visual representation helps confirm that x can be any number from 6 to infinity, making the solution clear and easy to understand.

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Most popular questions from this chapter

Concept Check Express each set in simplest interval form. \((-\infty,-1] \cap[-4, \infty)\)

Graph each linear inequality. $$ y \geq-\frac{3}{2} x+3 $$

Graph the solution set of each system of linear inequalities. $$ \begin{array}{r} x+y<4 \\ x-y>-4 \\ y>-1 \end{array} $$

Solve each compound inequality. Graph the solution set, and write it using interval notation. \(x+1>3\) or \(-4 x+1>5\)

Graph each linear inequality. $$ y \leq 4 x $$
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