The absolute value of a number can never be a negative number. Therefore, no value of \(\displaystyle x\) can make  \(\displaystyle | -3x - 15 | = -42\)  a true statement.

\(\displaystyle |x+3|=1\)

The equation involves an absolute value. First, we need to rewrite the equation with no absolute value.

\(\displaystyle x+3=\pm 1\)

We can split this equation into two possible equations.

Equation 1: \(\displaystyle x+3 =1\)

Equation 2: \(\displaystyle x+3 =-1\)

With two equations, there are two values for \(\displaystyle x\). Let's start with Equation 1.

\(\displaystyle x+3 =1\)

Subtract \(\displaystyle 3\) from both sides.

\(\displaystyle x+3-3=1-3\)

\(\displaystyle x=-2\)

That's the first value for \(\displaystyle x\). To get the second value for \(\displaystyle x\), we need to repeat the steps, but with Equation 2.

\(\displaystyle x+3 =-1\)

\(\displaystyle x+3-3=-1-3\)

\(\displaystyle x=-4\)

First, isolate the absolute value expression on one side.

\(\displaystyle \left | 2x-7 \right | -15 = -6\)

\(\displaystyle \left | 2x-7 \right | -15 + 15= -6 + 15\)

\(\displaystyle \left | 2x-7 \right | =9\)

Rewrite this as the compound statement:

\(\displaystyle 2x-7 =-9\)  or  \(\displaystyle 2x-7 =9\)

Solve each equation separately:

\(\displaystyle 2x-7 =-9\)

\(\displaystyle 2x-7+ 7 =-9 + 7\)

\(\displaystyle 2x = -2\)

\(\displaystyle 2x\div 2 = -2 \div 2\)

\(\displaystyle x=-1\)

\(\displaystyle 2x-7 =9\)

\(\displaystyle 2x-7+ 7 =9 + 7\)

\(\displaystyle 2x = 16\)

\(\displaystyle 2x\div 2 = 16 \div 2\)

\(\displaystyle x=8\)

Absolute value is a function that turns whatever is inside of it positive. This means that what's inside the function, \(\displaystyle 2x+1\), might be 7, or it could have also been -7. We have to solve for both situations.

a. \(\displaystyle 2x+1=7\) subtract 1 from both sides

\(\displaystyle 2x=6\) divide both sides by 2

\(\displaystyle x=3\)

b. \(\displaystyle 2x+1=-7\) subtract 1 from both sides

\(\displaystyle 2x=-8\) divide both sides by 2

\(\displaystyle x=-4\)

\(\displaystyle \\ \textup{We get rid of the absolute value by solving the two equations }\\3(x+1)^2=x+1\\\textup{and}\\3(x+1)^2=-(x+1)\\\textup{If } x\neq1\textup{ we can divide in both equations the two sides by }\\ x+1 \textup{ and obtain }\\ 3(x+1)=1 \\\textup{ and }}\\ 3(x+1)=-1 \\\textup{ which give}}\\ \textup{the solutions } x=-\frac{2}{3} \textup{ and } x=-\frac{4}{3}.\)

\(\displaystyle \textup{If } x=-1, \textup{both sides are equal to 0, so } -1 \textup{ is another solution.}\)

\(\displaystyle \left |3x-2 \right |=-4\)

First, ignore absolute value signs and solve for x.

\(\displaystyle 3x-2=-4\)

\(\displaystyle 3x=-2\)

\(\displaystyle x=-2/3\)

Now, for the other solution, ignore absolute value signs and change the sign of the right side of the equation. Solve for x.

\(\displaystyle 3x-2=4\)

\(\displaystyle 3x=6\)

\(\displaystyle x=2\)

First, make sure to have the absolute value part of the equation by itself on the left side.

\(\displaystyle \left | 4x\right |-2=10\)

\(\displaystyle \left | 4x\right |=12\)

Now solve for both x values.

\(\displaystyle 4x=12\)

\(\displaystyle x=3\)

Change the side of the right hand side to find the other value of x.

\(\displaystyle 4x=-12\)

\(\displaystyle x=-3\)

The question is asking for the value of \(\displaystyle x\). To isolate \(\displaystyle x\), we must first isolate the absolute value expression.

\(\displaystyle \left | 2x-3 \right | = 3\)

The absolute value bars give a positive value to the result of the expression within. This means the real value of the inner expression could be positive or negative. To find the two values of \(\displaystyle x\), create the two possible equations. 

\(\displaystyle 2x-3=3\)  \(\displaystyle \boldsymbol{\mathbf{OR}}\)  \(\displaystyle 2x-3=-3\)

Balance the equations to find \(\displaystyle x\).

\(\displaystyle 2x-3=3\)

\(\displaystyle 2x=6\)

\(\displaystyle x=3\)

--

\(\displaystyle 2x-3=-3\)

\(\displaystyle 2x=0\)

\(\displaystyle x=0\)

--

Therefore, \(\displaystyle x= 0\) and \(\displaystyle x=3\)

Also written as \(\displaystyle x= 0,3\).

When solving with absolute values, we need to consider both positive and negative answers.

\(\displaystyle x=10\)

\(\displaystyle -x=10\) 

Divide both sides by \(\displaystyle -1\), we get \(\displaystyle x=-10\).

Final answer is \(\displaystyle \pm10\).

When solving with absolute values, we need to consider both positive and negative answers.

\(\displaystyle x+5=2\) 

Subtract both sides by \(\displaystyle 5\), we get \(\displaystyle x=-3\). 

\(\displaystyle -(x+5)=2\) 

Distribute the negative sign to get \(\displaystyle -x-5=2\).

Add both sides by \(\displaystyle 5\) and divide both sides by \(\displaystyle -1\) to get \(\displaystyle x=-7\).

Final answer is \(\displaystyle -3, -7\).