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Chapter 2: Problem 37

Solve each of the inequalities and graph the solution set on a number line. \(3(x+2)>6\)

Short Answer

Expert verified
The solution is \(x > 0\), and it's graphed as an open circle at 0 with shading to the right.

Step by step solution

01

Distribute and Simplify

Start by distributing the 3 to both terms inside the parentheses in the inequality \(3(x + 2) > 6\). This gives us \(3x + 6 > 6\).
02

Subtract from Both Sides

Subtract 6 from both sides to isolate the term with x. So, \(3x + 6 - 6 > 6 - 6\), which simplifies to \(3x > 0\).
03

Divide to Solve for x

Divide each side of the inequality by 3 to solve for x. \(\frac{3x}{3} > \frac{0}{3}\), which results in \(x > 0\).
04

Graph the Solution on a Number Line

On a number line, draw an open circle at 0 to indicate that 0 is not included in the solution. Shade the region to the right of 0 to represent all numbers greater than 0.

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Key Concepts

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

Graphing Inequalities
Graphing inequalities on a number line is a great visual way to represent solution sets. When you have an inequality, such as \(x > 0\), it means that the solution includes all values of \(x\) that are greater than zero.
Once you've solved the inequality, the next step is to graph it, which can be achieved by following these steps:
  • Identify the critical value. In this case, it is 0.
  • Place an open circle at the critical value on the number line. An open circle tells us that the number itself is not part of the solution.
  • Shade the area on the number line that includes all potential solutions. For \(x > 0\), this will be everything to the right of the circle.
Graphing on a number line provides a clear and immediate understanding of where the values lie, making inequalities easy to interpret at a glance. This method also helps in comparing multiple inequalities visually.
Solving Inequalities
Solving inequalities involves finding the range of values that satisfy a given inequality statement. The process is similar to solving equations, but there are key differences to remember.

The process can be broken down into the following steps:
  • Begin by simplifying the inequality as you would with an equation. This includes distributing any multiplication over addition or subtraction and combining like terms.
  • Keep the inequality balanced. Whatever operation you perform on one side, you must do to the other side too.
  • Watch out for multiplication or division by a negative number. This flips the inequality symbol. For example, if you have \(-3x < 9\), dividing by \(-3\) will change it to \(x > -3\).
  • Once isolated, the inequality statement gives the solution set, which can be graphed for clarity.
Mastering solving inequalities will help you tackle a variety of problems in algebra and beyond, offering a way to express ranges of solutions rather than specific values.
Number Line
A number line is a straight line on which real numbers are represented as points. It's a fundamental tool for graphing inequalities and understanding number relationships. Here are some features and steps to use:
  • Zero is typically in the center, with positive numbers extending to the right and negative numbers stretching to the left.
  • It helps visualize solutions as continuums, enhancing comprehension especially for inequalities.
  • When graphing, placing open or closed circles and shading areas can easily show solutions.
Using number lines to represent inequalities allows students to physically see and better understand how numbers relate to each other in a solution set.

Understanding how to effectively utilize a number line is essential in solving mathematical problems, especially those involving inequalities, as it makes abstract concepts more concrete.

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Most popular questions from this chapter

For Problems \(31-54\), solve each inequality. $$ |x-3|>9 $$

For Problems \(35-44\), solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. $$ 3 x+2<-1 \text { or } 3 x+2>1 $$

Solve each inequality. $$ |3 x+1| \leq 13 $$

For Problems \(68-73\), solve each equation. $$ |3 x+1|=|2 x+3| $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -25 \leq 4 x+3 \leq 19 $$
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