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Chapter 7: Problem 2

Find the \(x\)-values that satisfy each statement. a. \(|x|=10\) b. \(|x|>4\)

Short Answer

Expert verified
a) \(x = 10\) or \(x = -10\); b) \(x > 4\) or \(x < -4\).

Step by step solution

01

Understanding Absolute Value Equation

To solve part (a), we need to understand that if \(|x|=10\), then \(x\) can be either 10 or -10, since both numbers have an absolute value of 10.
02

Solving the Equation \(|x|=10\)

Since \(|x| = 10\) is given, the solutions are \(x = 10\) and \(x = -10\). This is because the absolute value equation \(|x|=c\) means that \(x = c\) or \(x = -c\).
03

Understanding Absolute Value Inequality

For part (b), the inequality \(|x| > 4\) means that \(x\) is either greater than 4 or less than -4. This interpretation stems from the definition of absolute values in inequalities.
04

Solving the Inequality \(|x|>4\)

To solve \(|x| > 4\), we split it into two separate inequalities: \(x > 4\) or \(x < -4\). The solution is the union of these two intervals.

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Key Concepts

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

Understanding Inequalities with Absolute Values
Absolute value inequalities like \(|x| > 4\) tell us where a number is on the number line compared to the origin. The absolute value, \(|x|\), represents the distance of \(x\) from zero. For this inequality, we are looking for all values farther than 4 units from zero. This occurs in two parts of the number line:
  • To the right of 4 \((x > 4)\)
  • To the left of -4 \((x < -4)\)
Solving it requires splitting the inequality into these two parts. This concept is key because it shows how the absolute value creates a set interval on the real line, separating solutions into distinct regions.
Equation Solving with Absolute Values
Solving absolute value equations like \(|x|=10\) involves recognizing that absolute value measures 'distance' without regard to direction. Thus, if \(|x| = 10\), then \(x\) is 10 units away from zero, either to the right (positive) or to the left (negative). Therefore, the solutions are two possible values:
  • \(x = 10\) because moving 10 units to the right of zero is 10.
  • \(x = -10\) since moving 10 units to the left also yields -10.
This basic principle can be extended to solve more complex equations, showing that absolute value is about 'balancing' on both sides of a central point (here, zero).
Mathematical Reasoning in Solving Equations and Inequalities
Applying mathematical reasoning involves understanding the properties and behaviors of mathematical entities such as absolute values, equations, and inequalities. For example, knowing that absolute values represent distance is crucial in interpreting both \(|x| = c\) and \(|x| > c\). This reasoning helps translate mathematical language into graphical or number line representations. Key points of reasoning include:
  • Recognizing when absolute values form dual cases (positive and negative solutions).
  • Breaking down inequalities into separate comparisons (for greater or lesser solutions).
  • Using logic to determine solution sets and represent ranges on the number line.
Thus, reasoning allows us to navigate between different algebraic processes smoothly, structuring our approach to find solutions that satisfy mathematical statements.

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Most popular questions from this chapter

On graph paper, draw a graph that is not a function and has these three properties: Domain of \(x\)-values satisfying \(-3 \leq x \leq 5\) \- Range of \(y\)-values satisfying \(-4 \leq y \leq 4\) \- Includes the points \((-2,3)\) and \((3,-2)\)

For each relationship, identify the independent variable and the dependent variable. a. the weight of your dog and the reading on the scale b. the amount of time you spend in an airplane and the distance between your departure and your destination c. the number of times you dip a wick into hot wax and the diameter of a handmade candle

Solve each system of equations using the method of your choice. For each, tell which method you chose and why. a. \(\left\\{\begin{aligned}-2 x+3 y &=12 \\ 4 x-3 y &=-21 \end{aligned}\right.\) b. \(\left\\{\begin{array}{l}5 x+y=12 \\ 2 x-3 y=15\end{array}\right.\)

Find the slope of the line through each pair of points. a. \((1,3)\) and \((-2,6)\) (a) b. \((-4,-5)\) and \((7,0)\) c. \((-3,6)\) and \((9,6)\)

Solve each equation. a. \(2 x-5=7 x+15\) b. \(3(x+6)=12-5 x\) c. \(\frac{7(8-x)}{4}=x+3\)
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