Problem 65
Question
Solve each equation or inequality. $$4.2|0.5-x|+1=3.1$$
Step-by-Step Solution
Verified Answer
The solutions are \(x = 0\) and \(x = 1\).
1Step 1: Isolate the Absolute Value Expression
Start by isolating the absolute value expression \(|0.5 - x|\) in the equation. The given equation is:\[4.2|0.5-x|+1=3.1\]Subtract 1 from both sides to get:\[4.2|0.5-x| = 3.1 - 1\]\[4.2|0.5-x| = 2.1\]
2Step 2: Divide Both Sides by 4.2
Next, divide both sides of the equation by 4.2 to further simplify the equation:\[|0.5-x| = \frac{2.1}{4.2}\]\[|0.5-x| = 0.5\]
3Step 3: Break Down the Absolute Value Equation
The absolute value equation \(|0.5 - x| = 0.5\) can be broken down into two separate equations:1. \(0.5 - x = 0.5\)2. \(0.5 - x = -0.5\)
4Step 4: Solve the First Equation
Solve the first equation \(0.5 - x = 0.5\):Subtract 0.5 from both sides:\[-x = 0\]Multiply both sides by -1 to solve for \(x\):\[x = 0\]
5Step 5: Solve the Second Equation
Solve the second equation \(0.5 - x = -0.5\):Add \(x\) to both sides:\[0.5 = x - 0.5\]Add 0.5 to both sides:\[x = 1\]
6Step 6: Verify Solutions
Verify the solutions \(x = 0\) and \(x = 1\) by substituting them back into the original equation to ensure they satisfy the equation:1. For \(x = 0\): \[4.2|0.5 - 0| + 1 = 4.2 \times 0.5 + 1 = 3.1\] which is true.2. For \(x = 1\): \[4.2|0.5 - 1| + 1 = 4.2 \times 0.5 + 1 = 3.1\] which is true.
Key Concepts
Step-by-Step SolutionsEquation Solving TechniquesVerification of Solutions
Step-by-Step Solutions
In solving absolute value equations, breaking down the process into steps is crucial. Consider an equation like \[4.2|0.5-x|+1=3.1\]. First, isolate the absolute value term. This means getting rid of constants that are added or subtracted outside the absolute value. Here, subtracting 1 from both sides simplifies the equation to \(4.2|0.5-x|=2.1\).
Next, divide by the coefficient of the absolute value. This coefficient is 4.2, so dividing both sides by 4.2 gives \(|0.5-x| = 0.5\).
This demonstrates a key step in reducing absolute value equations to a simpler form. Reducing complexity step by step helps in breaking the absolute value into two possible equations. Thus, understanding these procedural steps is critical to finding the solution.
Next, divide by the coefficient of the absolute value. This coefficient is 4.2, so dividing both sides by 4.2 gives \(|0.5-x| = 0.5\).
This demonstrates a key step in reducing absolute value equations to a simpler form. Reducing complexity step by step helps in breaking the absolute value into two possible equations. Thus, understanding these procedural steps is critical to finding the solution.
Equation Solving Techniques
When facing an absolute value equation like \(|0.5-x| = 0.5\), it can be transformed into two different equations due to the nature of absolute values. Absolute value is essentially the distance a number is from zero, making it always non-negative. Thus, \(|A| = B\) is solved by creating:
By solving \(0.5 - x = 0.5\), subtract 0.5 from both sides leading to \(-x = 0\), or \(x = 0\) after multiplying by -1. The second equation, \(0.5-x = -0.5\), adds \(x\) then adds 0.5, giving \(x = 1\). Having these solutions, \(x=0\) and \(x=1\), highlights techniques of conversion and linear solving which are fundamental in algebra.
- \(A = B\)
- \(A = -B\)
By solving \(0.5 - x = 0.5\), subtract 0.5 from both sides leading to \(-x = 0\), or \(x = 0\) after multiplying by -1. The second equation, \(0.5-x = -0.5\), adds \(x\) then adds 0.5, giving \(x = 1\). Having these solutions, \(x=0\) and \(x=1\), highlights techniques of conversion and linear solving which are fundamental in algebra.
Verification of Solutions
Verification is an essential step and ensures accuracy. After finding potential solutions, always substitute them back into the original equation to confirm they work. For the solutions \(x = 0\) and \(x = 1\), substitute back into \(4.2|0.5-x|+1=3.1\).
When \(x = 0\), we substitute to get \(|0.5 - 0| = 0.5\), leading to calculations confirming \(4.2 \times 0.5 + 1 = 3.1\). Similarly, for \(x = 1\), substitution into the absolute value equation \(|0.5 - 1|\) results in \(0.5\) again, validating that \(4.2 \times 0.5 + 1 = 3.1\).
Both satisfy the original equation, thus verifying that both are indeed correct solutions. Verification ensures all of your hard work pays off and helps identify calculation errors if the class of solutions is mistaken.
When \(x = 0\), we substitute to get \(|0.5 - 0| = 0.5\), leading to calculations confirming \(4.2 \times 0.5 + 1 = 3.1\). Similarly, for \(x = 1\), substitution into the absolute value equation \(|0.5 - 1|\) results in \(0.5\) again, validating that \(4.2 \times 0.5 + 1 = 3.1\).
Both satisfy the original equation, thus verifying that both are indeed correct solutions. Verification ensures all of your hard work pays off and helps identify calculation errors if the class of solutions is mistaken.
Other exercises in this chapter
Problem 65
Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=3 x^{5}-x^{3}+7 x$$
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For certain pairs of functions \(f\) and \(g .(f \circ g)(x)=x\) and \((g \circ f)(x)=x\). Show that this is true for the pairs in Exercises \(65-68\). $$f(x)=4
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Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=x^{3}-4 x$$
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Solve each equation or inequality. $$|3 x-1|
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