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Chapter 6: Problem 29

Solve each inequality and graph the solution set on a number line. \(\frac{x}{3}>-2\)

Short Answer

Expert verified
The solution to the inequality is \(x>-6\). On a number line, this is represented by a circle at -6 (not filled in) and a line extending to the right of it.

Step by step solution

01

Isolate the Variable

To isolate x in the inequality \(\frac{x}{3}>-2\), multiply both sides by 3. This results in \(x>-6\).
02

Analyse the Solution

The inequality \(x>-6\) means that x can be any real number that is greater than -6.
03

Graph the Solution

On a number line, a circle should be drawn at -6 to represent that -6 is not included in the solution set (since the inequality is 'greater than' not 'greater than or equal to'). The circle should then have a line extending to the right of it, showing that any value larger than -6 is part of the solution set.

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Key Concepts

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

Graphing Inequalities
Graphing inequalities helps visualize the set of solutions. Begin by solving the inequality to express it in the standard form, like we did with \(x > -6\). Then, translate the solution onto the number line. This involves:
  • Marking a point where the inequality begins (e.g., -6 in our case).
  • Determining whether the point is part of the solution set, which it isn't here because -6 is not included.
  • Drawing an open circle to show that -6 is not part of the solution, and an arrow extending to the right to represent all numbers greater than -6.
The graph provides a quick way to check which numbers satisfy the inequality.
Solution Set
The solution set of an inequality refers to all the possible values that make the inequality true. For \(x > -6\), the solution set includes all numbers that are greater than -6. It's important to remember:
  • When an inequality uses the 'greater than' symbol (\(>\)), numbers larger but not including the given number are solutions.
  • Conversely, with 'greater than or equal to' (\(\geq\)), the specific number would be included in the solution set.
Visualizing the solution set as a collection of numbers helps in understanding and checking work.
Number Line
The number line is a tool for visual representation in math, especially helpful with inequalities. It allows for a clear illustration of the solution set. Here's how you work with it:
  • Identify the critical point; in \(x > -6\), that point is -6.
  • Use an open circle if the inequality is strictly greater/less than. Use a filled circle if it includes equality.
  • Draw a ray to show the direction of the solution set. For \(x > -6\), the ray points to the right.
Number lines make it much easier to interpret and verify solutions of inequalities.
Algebra Concepts
Understanding basic algebraic operations is key to solving inequalities. The rules for manipulating inequalities are similar to those for equations, with some specific distinctions:
  • Isolating the variable involves using inverse operations, such as multiplying the division \(\frac{x}{3} > -2\) by 3 to get \(x > -6\).
  • If you multiply or divide by a negative number, the inequality sign must be reversed.
  • Always perform the same operation to both sides of the inequality to maintain balance.
By mastering these algebra concepts, solving inequalities becomes straightforward and intuitive.

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Most popular questions from this chapter

Use FOIL to find the products in Exercises 1-8. \((3 x-7)(4 x-5)\)

It's easy to factor \(x^{2}+x+1\) because of the relatively small numbers for the constant term and the coefficient of \(x\).

Solve each equation by the method of your choice. \((2 x-6)(x+2)=5(x-1)-12\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}+4 x+1=0\)

Solve each equation using the zero-product principle. \((x+9)(3 x-1)=0\)
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