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Chapter 2: Problem 16

Solve. Graph the solution set and write it in interval notation. See Examples 2 through 4 $$ 7 x-1 \geq 6 x-1 $$

Short Answer

Expert verified
\( x \geq 0 \). Solution in interval notation: \([0, \infty)\).

Step by step solution

01

Isolate the Variable

We need to start by isolating the variable \( x \) in the inequality \( 7x - 1 \geq 6x - 1 \). To do this, subtract \( 6x \) from both sides of the inequality:\[7x - 6x - 1 \geq -1\]. Now simplify the expression to get \( x - 1 \geq -1 \).
02

Solve for x

Next, we need to isolate \( x \) by adding 1 to both sides of the inequality:\[x - 1 + 1 \geq -1 + 1\].This simplifies to \( x \geq 0 \). So, any value of \( x \) greater than or equal to 0 satisfies the inequality.
03

Graph the Solution Set

To graph the solution \( x \geq 0 \), draw a number line and shade the region to the right of 0, including the point at 0 itself, which is indicated by a closed dot at 0 because \( x \) can be equal to 0.
04

Write in Interval Notation

In interval notation, the solution set where \( x \geq 0 \) is written as:\[[0, \infty)\]. This means \( x \) starts from 0 and goes to positive infinity, including 0 but not infinity.

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Key Concepts

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

Graphing Inequalities
Graphing inequalities is a crucial skill to visualize solutions. When graphing, you first place a dot on the number line at the boundary point. This dot can be open or closed.
  • A **closed dot** (\( \bullet \)) on a number indicates that the number is included in the solution set. This happens in cases like \( x \geq 0 \), which means zero is part of the solutions.
  • An **open dot** (\( \circ \)) is used when the number is not part of the solution, for instance, in \( x > 0 \).
After placing the dot, shade the part of the number line that represents all solutions to the inequality. When the inequality is \( x \geq 0 \), shade the entire region to the right of zero to represent all numbers greater or equal to zero. Graphing helps provide a visual understanding of which numbers satisfy the inequality.
Interval Notation
Interval notation provides a concise way of expressing solution sets. Let's use \( x \geq 0 \) as an example.
  • Brackets **\([ ]\)** are used to include endpoint values when the inequality is "equal to".
  • Parentheses **\(( )\)** exclude endpoints from the solution set when the inequality is strictly greater or lesser.
In interval notation, the solution \( x \geq 0 \) is expressed as \([0, \infty)\). The square bracket includes 0, meaning zero is a solution, and the parenthesis at infinity indicates that infinity is not a specific number, so it's naturally not included.
Algebraic Manipulation
Algebraic manipulation is the process we undertake to alter equations or inequalities to find solutions. It's like solving a puzzle by rearranging pieces.
  • **Isolate the variable** to one side of the inequality to clearly see its relationship to other numbers. This usually starts with moving terms using addition or subtraction.
  • In the provided example, subtracting terms (\( 7x - 6x \)) helped simplify the original inequality \(7x - 1 \geq 6x - 1\) to \(x - 1 \geq -1\).
  • To simplify further, adjust the remaining terms to solve for the variable \( x \), such as by adding 1 to both sides of the simplified expression \( x - 1 \geq -1 \) to finally derive \( x \geq 0 \).
By sequentially isolating and solving incrementally for \( x \), algebraic manipulation clarifies the range of number values that satisfy the conditions of the inequality.

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Most popular questions from this chapter

Solve each inequality. Graph the solution set and write it in interval notation. $$ |y|>1 $$

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