Question

svlc algebra 1a - standard (15260)
solving absolute value equations
which statement is true?
the equation
-3|2x + 1.2| = -1
has no solution.
the equation
-0.3|3 + 8x| = 0.9
has no solution.
the equation
5|-3.1x + 6.9| = -3.5
has two solutions.
the equation
3.5|6x - 2| = 3.5
has one solution.

Explanation:

To determine which statement is true, we analyze each equation based on the properties of absolute value (the absolute value of a number is always non - negative, i.e., \(|a|\geq0\) for any real number \(a\)):

Analyze the first equation: \(- 3|2x + 1.2|=-1\)

Step 1: Isolate the absolute value expression

Divide both sides of the equation by \(-3\):
\(|2x + 1.2|=\frac{-1}{-3}=\frac{1}{3}\approx0.333\)
Since the absolute value of a number can be equal to a positive number, this equation has solutions.

Analyze the second equation: \(-0.3|3 + 8z| = 0.9\)

Step 1: Isolate the absolute value expression

Divide both sides by \(-0.3\):
\(|3 + 8z|=\frac{0.9}{-0.3}=- 3\)
But the absolute value of any real number is non - negative (\(|a|\geq0\) for all real \(a\)), and here we have \(|3 + 8z|=-3\) which is impossible. So this equation has no solution.

Analyze the third equation: \(5|-3.1x + 6.9|=-3.5\)

Step 1: Isolate the absolute value expression

Divide both sides by \(5\):
\(|-3.1x + 6.9|=\frac{-3.5}{5}=-0.7\)
Since the absolute value of a number is non - negative, and we have \(|-3.1x + 6.9|=-0.7\), this equation has no solutions (not two solutions).

Analyze the fourth equation: \(3.5|6x - 2| = 3.5\)

Step 1: Isolate the absolute value expression

Divide both sides by \(3.5\):
\(|6x - 2| = 1\)
This gives us two cases:
Case 1: \(6x-2 = 1\), then \(6x=1 + 2=3\), and \(x=\frac{3}{6}=0.5\)
Case 2: \(6x-2=-1\), then \(6x=-1 + 2 = 1\), and \(x=\frac{1}{6}\approx0.167\)
So this equation has two solutions, not one.

Answer:

The equation \(-0.3|3 + 8z| = 0.9\) has no solution.