Equations with Absolute Values: Absolute value on both sides of the equation

To solve the equation $|x+2|=|x-2|$, we begin by considering the properties of absolute values.

The statement $|A| = |B|$ implies two cases:

• Case 1: $A = B$
• Case 2: $A = -B$

For our problem, consider:

• Case 1: $x + 2 = x - 2$
• Case 2: $x + 2 = -(x - 2)$

Let's solve each case:

• Case 1: $x + 2 = x - 2$
  Subtract $x$ from both sides: $x + 2 - x = x - 2 - x$ Reduce to: $2 = -2$ Since $2 \neq -2$, this case has no solution.
• Case 2: $x + 2 = -(x - 2)$
  Expand the right-hand side: $x + 2 = -x + 2$ Add $x$ to both sides: $x + x + 2 = 2$ This simplifies to: $2x + 2 = 2$ Subtract 2 from both sides: $2x = 0$ Solve for $x$: $x = 0$

Thus, the solution to $|x+2|=|x-2|$ is $x = 0$. The correct answer is the choice: $x = 0$.

To solve the equation $|x-1| = |2x+3|$, follow these steps:

• First, recall the property: for any real numbers $a$ and $b$, $|a| = |b|$ implies $a = b$ or $a = -b$.
• We will consider two cases based on this property:

Case 1: Assume $x-1 = 2x+3$.
Simplify the equation:
$x-1 = 2x+3$
Subtract $x$ from both sides:
$-1 = x+3$
Subtract 3 from both sides:
$x = -4$.

Case 2: Assume $x-1 = -(2x+3)$.
Simplify the equation:
$x-1 = -2x-3$
Add $2x$ to both sides:
$3x-1 = -3$
Add 1 to both sides:
$3x = -2$
Divide everything by 3:
$x = -\frac{2}{3}$.

Therefore, the solutions to the equation $|x-1| = |2x+3|$ are $x = -4$ and $x = -\frac{2}{3}$.

These solutions correspond to answer choice 4: $x = -4$, $x = -\frac{2}{3}$.

Thus, $x = -4$ and $x = -\frac{2}{3}$.

To solve the problem, follow these steps:

• Case 1: Set $-x + 6 = 3x - 2$.
• Simplify and solve for $x$:

$-x + 6 = 3x - 2$
Add $x$ to both sides: $6 = 4x - 2$
Add 2 to both sides: $8 = 4x$
Divide both sides by 4: $x = 2$

• Case 2: Set $-x + 6 = -(3x - 2)$, which is $-x + 6 = -3x + 2$.
• Simplify and solve for $x$:

$-x + 6 = -3x + 2$
Add $3x$ to both sides: $2x + 6 = 2$
Subtract 6 from both sides: $2x = -4$
Divide both sides by 2: $x = -2$

Finally, verify that both solutions satisfy the original absolute value equation:

• When $x = 2$: $|-2 + 6| = |6 - 2| \Rightarrow |4| = |4|$, which holds true.
• When $x = -2$: $|2 + 6| = |-6 - 2| \Rightarrow |8| = |8|$, which holds true.

Thus, both $x = 2$ and $x = -2$ are valid solutions to the equation.

The solutions to the problem are $\boldsymbol{x=-2}$ and $\boldsymbol{x=2}$.

Therefore, the correct answer choice is $\boldsymbol{x=-2}$ , $\boldsymbol{x=2}$.

To solve the equation $|x+3| = |2x+6|$, we need to consider the properties of absolute values and analyze the cases for different ranges of $x$.

The equation $|x+3| = |2x+6|$ implies two possibilities based on absolute value properties:

• $x+3 = 2x+6$
• $x+3 = -(2x+6)$

Let's solve each case:

Case 1: $x+3 = 2x+6$

Simplify the equation:
$x + 3 = 2x + 6$
Move $x$ to the other side:
$3 = x + 6$
Subtract 6 from both sides:
$x = -3$

Case 2: $x+3 = -(2x+6)$

Simplify the equation:
$x+3 = -2x-6$
Add $2x$ to both sides:
$3x + 3 = -6$
Subtract 3 from both sides:
$3x = -9$
Divide both sides by 3:
$x = -3$

In both cases, we find that $x = -3$. However, we need to verify if $x = -3$ satisfies the original equation:

Substitute $x = -3$ into the original equation:
$|x+3| = |2x+6|$
$|-3+3| = |2(-3)+6|$
$|0| = |-6+6|$
$0 = 0$

Therefore, $x = -3$ satisfies the equation. The solution to the problem is $x = -3$.

In conclusion, the correct answer is $x = -3$.

To solve the given absolute value equation $|x+4| = |2x+20|$, we consider the properties of absolute values:

• If $|A| = |B|$, then $A = B$ or $A = -B$.

Applying this, we consider two cases:

• Case 1: $x+4 = 2x+20$

• Case 2: $x+4 = - (2x+20)$

Let's solve each case:

Case 1:

$x+4 = 2x+20$

Rearrange the equation:
$x+4 = 2x+20 \implies x - 2x = 20 - 4$
$-x = 16$
So, $x = -16$.

Case 2:

$x+4 = - (2x+20)$

Distribute the negative sign:
$x+4 = -2x-20$

Rearrange the equation:
$x + 2x = -20 - 4$
$3x = -24$
$x = -8$.

Therefore, the solutions to the equation are $x = -8$ and $x = -16$.

To verify, plug back $x = -8$ and $x = -16$ into the original equation:

• For $x = -16$: $|x+4| = |-16 + 4| = 12$ and $|2x+20| = |2(-16) + 20| = 12$. Both sides equal.

• For $x = -8$: $|x+4| = |-8 + 4| = 4$ and $|2x+20| = |2(-8) + 20| = 4$. Both sides equal.

Both solutions satisfy the original equation. Therefore, the correct answer is $x = -16$ and $x = -8$.

Thus, the solution to the absolute value equation is:

The solutions are $\mathbf{x = -8}$ and $\mathbf{x = -16}$.