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Chapter 9: Problem 65

Solve and graph the solution set on a number line. $$|x+6|>-10$$

Short Answer

Expert verified
The solution to the inequality \( |x + 6| > -10 \) is \( x < -16 \) and \( x > 4 \). When this is graphed on a number line, it entails all points to the left of -16 and all points to the right of 4, excluding -16 and 4, which have open circles.

Step by step solution

01

Understand absolute value inequalities

The first thing to understand about absolute value inequalities is that they can be split into two separate inequalities. The inequality \(|x+6|>-10\) says that the distance between x and -6 is greater than 10. Equivalently, x is either less than -6 - 10 = -16 or greater than -6 + 10 = 4.
02

Solve each inequality

So, we split the inequality into two inequalities: \(x + 6 < -10\) and \(x + 6 > 10\). Solving each inequality, we find \(x < -16\) and \(x > 4\). Note that since the inequality sign is '>', the solutions will not include -16 and 4.
03

Graph the solution set on a number line

On the number line, the solution to \(x < -16\) is all points to the left of -16 and the solution to \(x > 4\) is all points to the right of 4. Place open circles at -16 and 4 to indicate that these points are not 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.

Inequality Solutions
Dealing with absolute value inequalities involves working out the values for which the inequality holds true. An absolute value inequality like \(|x+6| > -10\) does not follow the usual pattern. Normally, with \(|x+a| > b\), we split it as either \(x+a > b\) or \(x+a < -b\). But here, the inequality is based on a concept that's always true because absolute values are non-negative. Thus, \(|x+6| > -10\) will hold true for all real numbers since absolute values can never be negative, let alone less than \(-10\). Remember:
  • If you encounter \(|expression| > negative\ value\), it's always true.
  • For any absolute inequality, break it into two inequalities to solve.
Graphing Inequalities on a Number Line
Visualizing solutions on a number line helps in understanding the scope of possible solutions. For \(x < -16\) or \(x > 4\), we graph this as follows:
  • Use a number line to represent all possible values of \(x\).
  • Draw open circles at \(-16\) and \(4\) because these exact points are not included in the solution.
  • Shade all values less than \(-16\) and all values greater than \(4\) since these are included.
The open circles indicate that the boundaries are not part of the solution. This is crucial to convey that \(x\) can be any number less than \(-16\) and greater than \(4\), excluding the points \(-16\) and \(4\) themselves.
Splitting Inequalities
A key method in solving absolute value inequalities is splitting them into two distinct inequalities. This is because absolute value represents distance. Here鈥檚 how it works:
  • Understand that \(|x+6| > -10\) implies that the expression inside the absolute value can take on values outside the usual expectations.
  • Absolute value always results in non-negative values, so any inequality compared to a negative number like \(-10\) is invariably true.
  • It's usual to split expressions like \(|x+a| < b\) or \(|x+a| > b\) to \(x+a < b\) and \(x+a > -b\).
The splitting approach allows for clear, manageable inequalities that are easier to solve by handling them separately. In traditional scenarios, this will guide you to find those specific boundary numbers.

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Most popular questions from this chapter

What is a system of linear inequalities?

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of nwo or more incqualities. By contrast, in Exercises \(53-54\) you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\)

The percentage, \(p,\) of defective products manufactured by a company is given by \(|p-0.3 \%| \leq 0.2 \% .\) If \(100,000\) products are manufactured and the company offers a \(\$ 5\) refund for each defective product, describe the company's cost for refunds.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Values of \(-5\) and 5 satisfy \(|x|=5,|x| \leq 5,\) and \(|x| \geq-5\)

Solve each inequality using a graphing utility. Graph each side separately in the same viewing rectangle. The solution set consists of all values of \(x\) for which the graph of the left side lies above the graph of the right side. $$|2 x-1|>7$$
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