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Chapter 7: Problem 4

Solve each equation for \(x\). Use a calculator graph to check your answers. a. \(|x-2|=4\) (a) b. \((x-2)^{2}=16\) c. \(|x+3|=7\) d. \((x+3)^{2}=49\)

Short Answer

Expert verified
a: \(x = 6, -2\); b: \(x = 6, -2\); c: \(x = 4, -10\); d: \(x = 4, -10\).

Step by step solution

01

Understanding the Absolute Value Equation

For equation a: \(|x-2| = 4\), we know that the absolute value equation \(|A| = B\) can be rewritten as two separate linear equations: \(A = B\) and \(A = -B\). Thus, we set up the following equations: \(x-2 = 4\) and \(x-2 = -4\).
02

Solving Each Linear Equation for x

For \(x - 2 = 4\), add 2 to both sides to get \(x = 6\). For \(x - 2 = -4\), add 2 to both sides to get \(x = -2\). Thus, the solutions are \(x = 6\) and \(x = -2\).
03

Understanding the Quadratic Equation

For equation b: \((x-2)^2 = 16\), we take the square root of both sides to remove the square term. Remember this results in two possible solutions: \(x-2 = \pm 4\).
04

Solving Each Resulting Equation

Solving \(x-2 = 4\) gives \(x = 6\) and solving \(x-2 = -4\) results in \(x = -2\). Hence, the solutions are \(x = 6\) and \(x = -2\).
05

Understanding the Absolute Value Equation

For equation c: \(|x+3| = 7\), we rewrite it into two equations: \(x+3 = 7\) and \(x+3 = -7\).
06

Solving Each Linear Equation for x

Solving \(x + 3 = 7\) gives \(x = 4\) and \(x + 3 = -7\) gives \(x = -10\). Thus, the solutions are \(x = 4\) and \(x = -10\).
07

Understanding the Quadratic Equation

For equation d: \((x+3)^2 =49\), we take the square root of both sides: \(x+3 = \pm 7\) to solve for x.
08

Solving Each Resulting Equation

For \(x + 3 = 7\), subtract 3 from both sides to get \(x = 4\). For \(x + 3 = -7\), subtract 3 from both sides to get \(x = -10\). Thus, the solutions are \(x = 4\) and \(x = -10\).
09

Check Solutions Using Calculators or Graphs

Plot each equation on a graph calculator or graphing tool to visually confirm the intersection points, which should match the x-values from our solutions.

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Key Concepts

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

Absolute Value Equations
Absolute value equations can be a bit puzzling at first. The absolute value of a number is its distance from zero on a number line, irrespective of direction. That's why equations like \(|x-2| = 4\) result in two separate scenarios: \(x-2 = 4\) and \(x-2 = -4\). We essentially have to consider both the positive and the negative distances.
For solving these, simply isolate \(x\) in each equation.
  • When solving \(x - 2 = 4\): Add 2 to both sides to find \(x = 6\).
  • When solving \(x - 2 = -4\): Add 2 to both sides to find \(x = -2\).
This gives us two solutions: \(x = 6\) and \(x = -2\). You're always looking for these pairs when solving absolute value equations. Always remember: the absolute value creates two diverging paths, reflecting both positive and negative counterparts.
Quadratic Equations
Quadratic equations are fascinating because they introduce us to the world of squares and roots. Consider the equation \((x-2)^2 = 16\). The primary aim is to eliminate the square on \((x-2)\) by taking the square root of both sides.
  • Take the square root: \(x-2 = \pm 4\). This shows two possible solutions because both \(4^2\) and \((-4)^2\) equal 16.
  • Solving \(x-2 = 4\): Add 2 gives \(x = 6\).
  • Solving \(x-2 = -4\): Add 2 gives \(x = -2\).
Now, we have both results: \(x = 6\) and \(x = -2\). The square dictates two outcomes each time, making quadratic equations unique compared to linear ones.
Solving Equations
Solving equations effectively revolves around understanding and applying specific rules for different types. Be it absolute value or quadratic equations, the steps involve isolating \(x\) by a series of logical reversals.
For simplicity:
  • Identify the type: Whether it's absolute value or quadratic affects the approach.
  • Break it down: For absolute values, break into two linear equations; for quadratics, take the square root.
  • Step-by-step solving: Methodically solve each resulting equation for \(x\).
  • Verification: Substitute the solutions back in to check their correctness.
The key to success is recognizing the nature of the equation and following logical steps to dismantle it systematically.
Graphical Solutions
Graphical solutions are an excellent way to visually confirm the results of algebraic equations. By plotting the equation on a graphing calculator or tool, you can visually see the intersections. These points represent potential solutions where the algebraic method found them.
  • Advantages: Visual confirmations provide a powerful check against computation mistakes.
  • Insightful: Understanding equations graphically can increase comprehension of concepts like intersection points and symmetry.
  • Practical: Helps in scenarios where algebraic solutions are complex or error-prone.
Graphs turn abstract numbers into concrete visuals and make the theoretical world of equations come alive with intuitive clarity.

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Most popular questions from this chapter

For what values of \(y\) does the equation \(y=x^{2}\) have a. No real solutions? b. Only one solution? c. Two solutions?

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

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

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Use properties of exponents to find an equivalent expression in the form \(a x^{n}\), if possible. Use positive exponents. a. \(24 x^{6} \cdot 2 x^{3}\) (a) b. \(\left(-15 x^{4}\right)\left(-2 x^{4}\right)\) c. \(\frac{72 x^{11}}{3 x^{2}}\) d. \(4 x^{2}\left(2.5 x^{4}\right)^{3}\) e. \(\frac{15 x^{5}}{-6 x^{2}}\) f. \(\left(-3 x^{3}\right)\left(4 x^{4}\right)^{2}\) g. \(\frac{42 x^{-6} y^{2}}{7 y^{-4}}\) h. \(3\left(5 x y^{2}\right)^{3}\)
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