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Chapter 6: Problem 46

Solve each inequality and graph the solution set on a number line. \(3-x \geq-3\)

Short Answer

Expert verified
The solution for the inequality \(3-x \geq -3\) is \(x \leq 6\).

Step by step solution

01

Rewrite inequality

Rewrite the inequality \(3 - x \geq -3\) as \(-x \geq -3 - 3\). This formulates equivalent inequality.
02

Simplify Right Hand Side

Simplify the right-hand side of the inequality, resulting in \(-x \geq -6\).
03

Divide by Negative Value

Divide both sides of the inequality by -1. But remember, dividing by a negative number flips the direction of the inequality. So it becomes \(x \leq 6\).
04

Plot the solution on a number line

Graph the solution on a number line. This will include all values less than or equal to 6.

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Key Concepts

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

Solve Inequalities
Inequalities are just like equations but with inequality signs such as \(<, \leq, >, \geq\). Solving inequalities involves finding the range of values for which the inequality is true.
To solve an inequality, follow similar steps as solving equations:
  • Isolate the variable on one side of the inequality sign.
  • Perform algebraic operations such as addition, subtraction, multiplication, or division.
  • Pay special attention when multiplying or dividing by a negative number, as this action reverses the inequality sign.
Remember, the solution to an inequality is often a range of values, not just a single number.
Inequalities on a Number Line
A number line is a visual way to represent solutions of inequalities. It helps in understanding which numbers satisfy the inequality.
Here's how to depict inequalities on a number line:
  • Identify the solution from the inequality statement.
  • Draw a number line, marking the critical point where the inequality shifts direction.
  • If the inequality includes equal to (\(\leq\) or \(\geq\)), use a closed dot on the number line at that point; otherwise, use an open dot (\(<\) or \(>\)).
  • Shade the region on the number line that represents all the solutions to the inequality.
By graphing on a number line, it becomes clear which values are included in the solution set.
Inequality Solution Steps
The process of solving an inequality can be broken down into a series of logical steps. These steps ensure accuracy and comprehension.For example, with the original inequality: \[3 - x \geq -3\] The steps to solve would be:
  • Step 1: Rewrite Inequality
    Substitute terms to get the unknown on one side: \(-x \geq -3 - 3\).
  • Step 2: Simplify
    Combine like terms: \(-x \geq -6\).
  • Step 3: Handle Negative Coefficient
    When you divide both sides by \(-1\), flip the inequality sign: \(x \leq 6\).
Having a structured approach with these steps makes tackling inequalities more straightforward.
Graphing Inequalities
Graphing inequalities is a fundamental tool for understanding and visualizing solutions. This technique allows us to see all possible solutions in one glance.
To graph the inequality \(x \leq 6\) on a number line:
  • Draw a horizontal line and label it with numbers, including the important values surrounding 6.
  • Because our inequality is \(\leq\), place a filled dot (or circle) on the number 6, indicating that 6 is included in the solution set.
  • Shade all the numbers to the left of 6, showing that all those numbers are part of the solution.
Graphing the inequality visually communicates which numbers satisfy the condition, making it a vital step in solving and understanding inequalities.

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Most popular questions from this chapter

The formula $$ N=\frac{t^{2}-t}{2} $$ describes the number of football games, \(N\), that must be played in a league with t teams if each team is to play every other team once. Use this information to solve Exercises 83-84. If a league has 45 games scheduled, how many teams belong to the league, assuming that each team plays every other team once?

The radicand of the quadratic formula, \(b^{2}-4 a c\), can be used to determine whether \(a x^{2}+b x+c=0\) has solutions that are rational, irrational, or not real numbers. Explain how this works. Is it possible to determine the kinds of answers that one will obtain to a quadratic equation without actually solving the equation? Explain.

Factor the trinomials or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(x^{2}+17 x+16\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}+5 x+2=0\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}-2 x-10=0\)
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