Which statement is true?
The equation $-3|2 x+1.2|=-1$ has no solution.
The equation $3.5|6 x-2|=3.5$ has one solution.
The equation $5|-3.1 x+6.9|=-3.5$ has two solutions.
The equation $-0.3|3+8 x|=0.9$ has no solution.
Answer (2)
Equation − 3∣2 x + 1.2∣ = − 1 has two solutions.
Equation 3.5∣6 x − 2∣ = 3.5 has two solutions.
Equation 5∣ − 3.1 x + 6.9∣ = − 3.5 has no solution.
Equation − 0.3∣3 + 8 x ∣ = 0.9 has no solution. Thus, the true statement is: The equation − 0.3∣3 + 8 x ∣ = 0.9 has no solution. T r u e
Explanation
Understanding the Problem We are given four equations involving absolute values and asked to determine which statement about the number of solutions is true.
Analyzing Equation 1 Equation 1: − 3∣2 x + 1.2∣ = − 1 . We need to isolate the absolute value.
Solving Equation 1 Divide both sides of Equation 1 by -3: ∣2 x + 1.2∣ = − 3 − 1 = 3 1 . Since the absolute value is equal to a positive number, this equation has two solutions.
Analyzing Equation 2 Equation 2: 3.5∣6 x − 2∣ = 3.5 . We need to isolate the absolute value.
Solving Equation 2 Divide both sides of Equation 2 by 3.5: ∣6 x − 2∣ = 3.5 3.5 = 1 . Since the absolute value is equal to a positive number, this equation has two solutions.
Analyzing Equation 3 Equation 3: 5∣ − 3.1 x + 6.9∣ = − 3.5 . We need to isolate the absolute value.
Solving Equation 3 Divide both sides of Equation 3 by 5: ∣ − 3.1 x + 6.9∣ = 5 − 3.5 = − 0.7 . Since the absolute value of any expression must be non-negative, this equation has no solution.
Analyzing Equation 4 Equation 4: − 0.3∣3 + 8 x ∣ = 0.9 . We need to isolate the absolute value.
Solving Equation 4 Divide both sides of Equation 4 by -0.3: ∣3 + 8 x ∣ = − 0.3 0.9 = − 3 . Since the absolute value of any expression must be non-negative, this equation has no solution.
Conclusion Based on the analysis above:
Equation 1 has two solutions.
Equation 2 has two solutions.
Equation 3 has no solution.
Equation 4 has no solution. Therefore, the statement 'The equation − 0.3∣3 + 8 x ∣ = 0.9 has no solution' is true.
Examples
Absolute value equations are useful in many real-world scenarios, such as calculating tolerances in manufacturing. For example, if a machine is supposed to cut a metal rod to a length of 10 cm with a tolerance of 0.1 cm, the actual length, x , must satisfy the equation ∣ x − 10∣ l e 0.1 . This ensures that the rod is within acceptable limits. Similarly, in finance, absolute values can be used to model deviations from expected returns on investments.
The true statement is that the equation − 0.3∣3 + 8 x ∣ = 0.9 has no solution. The other equations either have two solutions or are not solvable due to resulting in negative values for absolute expressions. Therefore, the correct answer is the fourth statement.
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