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Chapter 7: Problem 8

Solve each equation for \(x\). a. \(|x|=12\) b. \(10=|x|+4\) c. \(10=2|x|+6\) d. \(4=2(|x|+2)\)

Short Answer

Expert verified
a. \(x = 12, -12\), b. \(x = 6, -6\), c. \(x = 2, -2\), d. \(x = 0\)

Step by step solution

01

Understand Absolute Value

The absolute value of a number is its distance from zero on the number line. This means that both \(|x| = a\) and \(|x| = -a\) yield the same result as the absolute value must always be positive. Therefore, an equation involving an absolute value will typically have two solutions: one positive and one negative.
02

Solve Equation a: \(|x| = 12\)

For \(|x| = 12\), this means \(x = 12\) or \(x = -12\) because \(|x|\) is the same as the distance from zero, either positive or negative.
03

Solve Equation b: \(10 = |x| + 4\)

First, isolate the absolute value by subtracting 4 from both sides: \(|x| = 6\).Now, recall for absolute values, \(x = 6\) or \(x = -6\).
04

Solve Equation c: \(10 = 2|x| + 6\)

Subtract 6 from both sides: \(4 = 2|x|\).Next, divide both sides by 2 to isolate the absolute value: \(|x| = 2\).Now, \(x = 2\) or \(x = -2\).
05

Solve Equation d: \(4 = 2(|x| + 2)\)

First, distribute the 2: \(4 = 2|x| + 4\).Subtract 4 from both sides to simplify: \(0 = 2|x|\).Divide both sides by 2: \(|x| = 0\).Therefore, \(x = 0\).

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Key Concepts

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

Absolute Value
Absolute value refers to how far a number is from zero on the number line, regardless of direction. It's like saying, "Forget if it鈥檚 left or right, just tell me the distance!"
Imagine you drop a basketball from 10 feet; it bounces up and down. You don't care what direction it's bouncing; you just want to know how high it bounces each time (its absolute value).
Key things about absolute value:
  • Absolute value is always positive or zero.
  • It represents distance and is shown as \(|x|\).
  • In equations, solve by considering both directions (positive and negative).
When you see \(|x| = a\), think: \(|x|\) can equal 'a' or '-a'. This is critical for finding both solutions of an equation!
Distance from Zero
Distance from zero is a simple idea, yet it forms the backbone of understanding absolute values. Every number on the number line has a certain distance from the zero point, and this distance is its absolute value.
For instance, the numbers 5 and -5 each have an absolute value of 5 because they are both 5 units away from zero. The same goes for absolute equations like \(|x| = 12\). Here, the number 'x' can be 12 or -12.
This "double" solution happens because:
  • +12 is 12 steps to the right of zero.
  • -12 is 12 steps to the left of zero.
In essence, the main idea is that the direction towards zero doesn't matter. Distance from zero helps visualize equations that involve absolute values.
Isolating Variables
Isolating variables is like untangling a knot to find 'x' all alone on one side of the equation. With absolute equations, it's often the first goal.
To isolate, you need to strip away everything sticking to the \(|x|\) part.

For example, in the equation \(10 = |x| + 4\):
  • First step: remove the 4 by subtracting it from both sides, which leaves: \( |x| = 6\).
Even for trickier ones like \(10 = 2|x| + 6\), here is the process:
  • Subtract 6 from both sides: \(4 = 2|x|\).
  • Then divide everything by 2 to find \(|x|\), making it: \( |x| = 2\).
Once isolated, proceed to interpret the solutions of the absolute value.
Positive and Negative Solutions
Equations involving absolute values usually have two solutions, resembling the scenario "two sides of a coin".
Once the absolute value is isolated, it's essential to remember both the positive and negative potential solutions.

Take \(|x| = 6\):
  • This means 'x' could either be 6 or -6. Both satisfy the equation.
For each solution: consider two angles
  • Positive: The straightforward answer, akin to solving 'x' as it appears.
  • Negative: Don鈥檛 forget, the negative reflection is also valid!
In sum, solving these equals more than one attempt and more thorough results because you're addressing every possibility鈥攋ust like ensuring to find all zeroes in a detailed number game.

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Most popular questions from this chapter

The solutions to the equation \(|x-4|+3=17\) are \(-10\) and 18 . a. Explain why the equation has two solutions. b. What are the solutions to \(|x-4|+3 \leq 17\) ? Explain. (i) c. What are the solutions to \(|x-4|+3>17\) ? Explain.

Solve each system of equations using the method of your choice. For each, tell which method you chose and why. a. \(|x+1|=7\) (A) b. \(2|3 x-1|=4\) c. \(|2 x-4.2|-3=-3\) d. \(3|x+2|=-6\)

Sketch a graph of a continuous function to fit each description. a. always increasing with a faster and faster rate of change b. decreasing with a slower and slower rate of change, then increasing with a faster and faster rate of change (a) c. linear and decreasing d. decreasing with a faster and faster rate of change

Write an equation for the function represented in each table. Use your calculator to check your answers. $$ \begin{aligned} &\text { a. }\\\ &\begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & -3 & -1 & 0 & 1 & 4 & 6 \\ \hline \boldsymbol{y} & 14 & 10 & 8 & 6 & 0 & -4 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { b. }\\\ &\begin{array}{|c|c|c|c|c|c|c|} \hline x & -3 & -1 & 0 & 1 & 4 & 6 \\ \hline y & 9 & 1 & 0 & 1 & 16 & 36 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { c. }\\\ &\begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & -3 & -1 & 0 & 1 & 4 & 6 \\ \hline \boldsymbol{y} & 3 & 1 & 0 & 1 & 4 & 6 \\ \hline \end{array} \end{aligned} $$

This 4-by-4 grid contains squares of different sizes. a. How many of each size square are there? Include overlapping squares. b. How many total squares would a 3 -by-3 grid contain? A 2 -by-2 grid? A l-by-1 grid? c. Find a pattern to determine how many squares an \(n\)-by- \(n\) grid contains. Use your pattern to predict the number of squares in a 5-by-5 grid.
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