Question

drag the tiles to the correct boxes to complete the pairs.
how many solutions does each of these equations have?
one solution two solutions no solutions
|2x + 1| = 0
|x + 21| = 2
2|x - 2| = -1

Explanation:

for \(|2x + 1| = 0\):

Step1: Recall absolute value property

The absolute value equation \(|A| = 0\) implies \(A = 0\) (since the only number with absolute value 0 is 0 itself). So for \(|2x + 1| = 0\), we set \(2x+1 = 0\).

Step2: Solve for \(x\)

\(2x+1 = 0\)
Subtract 1 from both sides: \(2x= - 1\)
Divide by 2: \(x =-\frac{1}{2}\)
This gives one solution.

for \(|x + 21| = 2\):

Step1: Recall absolute value property

The absolute value equation \(|A| = B\) (where \(B>0\)) implies \(A = B\) or \(A=-B\). Here \(B = 2>0\), so for \(|x + 21| = 2\), we have two cases: \(x + 21=2\) or \(x + 21=-2\).

Step2: Solve each case

• Case 1: \(x + 21=2\)

Subtract 21 from both sides: \(x=2 - 21=-19\)

• Case 2: \(x + 21=-2\)

Subtract 21 from both sides: \(x=-2 - 21=-23\)
This gives two solutions.

for \(2|x - 2| = -1\):

Step1: Simplify the equation

Divide both sides by 2: \(|x - 2|=-\frac{1}{2}\)

Step2: Recall absolute value property

The absolute value of a number is always non - negative (i.e., \(|A|\geq0\) for any real number \(A\)). But here we have \(|x - 2|=-\frac{1}{2}\), and \(-\frac{1}{2}<0\). Since the absolute value can never be negative, there are no solutions.

Answer:

• \(|2x + 1| = 0\) \(

ightarrow\) one solution

• \(|x + 21| = 2\) \(

ightarrow\) two solutions

• \(2|x - 2| = -1\) \(

ightarrow\) no solutions

So the pairings are:
\(|2x + 1| = 0\) \(
ightarrow\) one solution
\(|x + 21| = 2\) \(
ightarrow\) two solutions
\(2|x - 2| = -1\) \(
ightarrow\) no solutions