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Chapter 9: Problem 34

Solve. $$\left|\frac{5}{4} k+2\right|+9=7$$

Short Answer

Expert verified
The given equation has no solution because the absolute value of any number is always nonnegative, and the right-hand side of the isolated equation is -2, which is negative. Therefore, there is no value of k that will satisfy the equation.

Step by step solution

01

Isolate the absolute value term

Subtract 9 from both sides of the equation to isolate the absolute value term: \[\left|\frac{5}{4} k+2\right| = -2\]
02

Check the feasibility of the solution

Since the absolute value of any number is always nonnegative, this equation has no solution because the right-hand side is negative (-2). Thus, there is no value of k that will satisfy the equation. If you want to make sure, you can further analyze the equation by breaking it into two cases.
03

Create two cases (not necessary for this specific problem, but for understanding purposes)

Case 1 - The expression inside the absolute value is nonnegative: \(\frac{5}{4} k + 2 \ge 0\), so we have: \(\frac{5}{4} k + 2 = -2\) Solve this equation for k, which results in: \(k = \frac{-5}{2}\) However, plugging this back into the inequality we have: \(\frac{5}{4} \left(\frac{-5}{2}\right) + 2 = -\frac{5}{2}\) This contradict the assumption that the expression inside the absolute value should be nonnegative, so there is no solution in this case. Case 2 - The expression inside the absolute value is negative: \(\frac{5}{4} k + 2 < 0\), so we have: \(-\left(\frac{5}{4} k + 2\right) = -2\) Solve this equation for k, which results in: \(k =\frac{5}{2}\) However, plugging this back into the inequality we have: \(\frac{5}{4} \left(\frac{5}{2}\right) + 2 = \frac{5}{2}\) This also contradicts the assumption that the expression inside the absolute value should be negative, so there is no solution in this case. Since there are no valid solutions in either case, we conclude that the given equation has no solutions.

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions form the building blocks of equations. They are combinations of numbers, variables, and arithmetic operations. In our problem, we encounter the expression \( \frac{5}{4} k + 2 \). Here, \( k \) is a variable, and \( \frac{5}{4} \) and \( 2 \) are constants.

By understanding how these components come together, we can manipulate them to solve equations. Remember:
  • Variables, like \( k \), represent unknown values.
  • Constants, like \( \frac{5}{4} \), are fixed values.
  • Operations, like addition and multiplication, dictate how these parts interact.
Grasping how these elements interact is key to solving the entire equation.
Techniques for Solving Equations
Solving equations often requires isolating the variable to one side of the equation. To do this, we might use inverse operations like subtraction, addition, multiplication, or division. In the initial step of solving \( \left|\frac{5}{4} k+2\right|+9=7 \), we isolate \( \left|\frac{5}{4} k+2\right| \) by subtracting 9 from both sides.

Once isolated, we must recognize the peculiarity of absolute value equations which can split into two distinct cases, positive and negative, akin to checking dual scenarios. This step isn't essential here due to the negative result, but it鈥檚 helpful for other equations:
  • The inside of the absolute value is set equal to the positive counterpart.
  • It's also set to the negative of the counterpart.
These techniques help us explore all potential solutions, providing a comprehensive method for tackling various equations.
No Solution Scenarios in Equations
An equation might have no solution when the calculations contradict the fundamentals of mathematics, especially with absolute values. An absolute value represents the distance from zero on a number line, which is always nonnegative.

In our case, \( \left|\frac{5}{4} k+2\right| = -2 \) defies logic, as the result should be nonnegative. When faced with such contradictions:
  • Always check the validity of the outcome against mathematical principles.
  • Rely on reasoned assumptions, such as acknowledging absolute values cannot be negative.
Recognizing scenarios where there are no possible solutions is crucial in problem-solving, saving time and ensuring a deeper understanding of mathematical concepts.

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Most popular questions from this chapter

Solve each inequality. Graph the solution set and write the answer in interval notation. $$-3+\left|\frac{5}{6} n+\frac{1}{2}\right| \geq 1$$

The graphs of compound linear inequalities in two variables are given next. For each, find three points that are in the solution set and three that are not. \(x>4\) and \(y \leq-\frac{2}{3} x+2\)

The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$|7 z-8| \leq 0$$

The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$-\frac{3}{5} \geq \frac{5}{2} a-\frac{1}{2}$$

Solve each inequality. Graph the solution set and write the answer in interval notation. $$\left|\frac{3}{2} y-\frac{5}{4}\right|+9 \geq 11$$
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