Solve for $\displaystyle x$ given $\displaystyle |7x+14+22x|-72=-3x$ 

When given an absolute value equation recognize there are often multiple solutions. The reason why is best exhibited in a simpler example:

Given $\displaystyle |x|=1$  solve for values $\displaystyle X$ of that make this statement true. When you taken an absolute value of something you always end up with the positive number so both $\displaystyle -1$ and $\displaystyle 1$ would make this statement true. The solutions can also be written as ±$\displaystyle 1$.

In the case of the more complicated equation $\displaystyle |7x+14+22x|-72=-3x$, however before proceeding let's simplify this equation a little as there are two $\displaystyle x$ terms that can be added together within the absolute value. These are $\displaystyle 7x$ and $\displaystyle 22x$.

When added together this gives $\displaystyle 29x$,

thereby giving you $\displaystyle |29x+14|-72=-3x$

For the same reason as shown in the case of $\displaystyle |x|=1$  there are potentially two solutions to this equation, which are shown by $\displaystyle ±(29x+14)-72=-3x$ as an absolute value will always end up creating a positive result. Make sure you apply the parentheses to only the portion of the equation with the absolute value otherwise your answer will be incorrect.

To simplify the absolute value we must look at each of these cases:

 Let's start with the positive case:

$\displaystyle (29x+14)-72=-3x$

Just like a normal equation with one unknown we will simplify it by isolating $\displaystyle x$ by itself. This is first done by combining like terms and getting $\displaystyle x$ on one side of the equation.

We can first subtract $\displaystyle 72$ from $\displaystyle 14$ which is $\displaystyle -58$, this gives you:

$\displaystyle 29x-58=-3x$

Next we can subtract $\displaystyle 29x$ from both sides of the equation.

$\displaystyle -58=-26x$

Dividing both sides by $\displaystyle -26$ gives you a final answer of $\displaystyle x=58/26$ this however can be simplified to $\displaystyle 29/16$  as both the numerator and denominator are divisible by $\displaystyle 2$.

 so  $\displaystyle x=29/16$

To check this solution it must be substituted in the original absolute value for $\displaystyle x$ and if it's a correct answer you'll end up with a true statement, so

$\displaystyle |29x+14|-72=-3x$,$\displaystyle x=29/16$

$\displaystyle |29(29/16)+14|-72=-3(29/16)$

Simplify the equation by multiplying $\displaystyle 29$ by $\displaystyle 29/16$ and $\displaystyle -3$ by $\displaystyle 29/16$

$\displaystyle 29*29/16=841/16$

$\displaystyle -3*29/16=-87/16$

This leaves our equation with

$\displaystyle |841/16+14|-72=-87/16$ 

Next add $\displaystyle 72$ to both sides of the equation:

$\displaystyle |841/16+14|=72-87/16$

In order to simplify $\displaystyle 72-87/16$  you must find a common denominator, which is most easily done by multiplying $\displaystyle 72*16/16$.

This leaves:

$\displaystyle |841/16+14|=1152/16-87/16$

$\displaystyle |841/16+14|=1065/16$

Similarly a common denominator is found for $\displaystyle 841/16$  and $\displaystyle 14$ by multiplying $\displaystyle 14$ by  $\displaystyle 16/16$ which gives you $\displaystyle 224/16$

this leaves:

$\displaystyle |841/16+14*16/16|=1065/16$

$\displaystyle |841/16+224/16|=1065/6$

simplifying further gives you 

$\displaystyle |1065/6|=1065/6$ which is a true statement, so $\displaystyle x=29/16$ is a valid solution

Next let's solve for the negative case:

$\displaystyle -(29x+14)-72=-3x$

Distribute the negative sign, which is just $\displaystyle -1$ to make calculations easier and you'll get:

$\displaystyle -29x-14-72=-3x$ 

Combine like terms:

$\displaystyle -86=26x$

$\displaystyle x=-86/26$

This can be simplified to $\displaystyle -43/13$ as both the numerator and denominator are divisible by $\displaystyle 2$, therefore you final answer is $\displaystyle x=-43/13$

Checking this solution is done just as you did for the previous solution obtained.

Given $\displaystyle |29x+14|-72=-3x,$$\displaystyle x=-43/13$

substitute $\displaystyle -43/13$ in for $\displaystyle x$

$\displaystyle |29(-43/13)+14|-72=-3(-43/13)$

Multiplying $\displaystyle 29$ by $\displaystyle (-43/13)$ gives $\displaystyle -1247/13$

Multiplying $\displaystyle -3*(-43/13)$ gives $\displaystyle 129/13$

So the equation simplifies to:

$\displaystyle |-1247/13+14|-72=129/13$

Next $\displaystyle 72$ can be added to both sides of the equation giving:

$\displaystyle |-1247/13+14|=129/13+72$ 

Now common denominators must be found for $\displaystyle 129/13+72$ and $\displaystyle -1247/13+14.$

The common denominator for $\displaystyle 129/13$ is found by multiplying $\displaystyle 72$ by $\displaystyle 13/13$. This gives you $\displaystyle 936/13+129/13$. This can then be simplified through addition and gives $\displaystyle 1065/13$.

The common denominator of $\displaystyle -1247/13+14$ is found by multiplying $\displaystyle -1247/13*14/14$ and $\displaystyle 14*(13*14)/(13*14)$

$\displaystyle -1247/13*14/14=-17458/182$

$\displaystyle 14*(13*14)/(13*14)=14*182/182=2548/182$

so$\displaystyle -17458/182+2548/182=-14910/182$ 

The simplified equation becomes

$\displaystyle |-19410/182|=1065/13$

Through dividing $\displaystyle -19410/182$ by $\displaystyle 13$ you get $\displaystyle -1065/13$ and the absolute value of $\displaystyle -19410/182$  is $\displaystyle 1065/13$, so you get $\displaystyle 1065/13$. This is true statement

so $\displaystyle x=-43/13$ is also a valid solution.

To solve absolute value equations, we must understand that the absoute value function makes a value positive. So when we are solving these problems, we must consider two scenarios, one where the value is positive and one where the value is negative.

$\displaystyle \left | 2x-5\right |=3x$

$\displaystyle 2x-5=3x$ 

and 

$\displaystyle 2x-5=-(3x)$

This gives us:

$\displaystyle -5=x$ and $\displaystyle 5x=5$

$\displaystyle x=-5\ and\ 1$

However, this question has an $\displaystyle x$ outside of the absolute value expression, in this case $\displaystyle 3x$. Thus, any negative value of $\displaystyle x$ will make the right side of the equation equal to a negative number, which cannot be true for an absolute value expression. Thus, $\displaystyle x=-5$ is an extraneous solution, as $\displaystyle \left | 2x-5\right |$ cannot equal a negative number.

Our final solution is then

$\displaystyle x=1$

$\displaystyle \small \left | x-4 \right | \geq 7$

$\displaystyle \small x-4\geq7\ or\ x-4\leq-7$

$\displaystyle \small x\geq11\ or\ x\leq-3$

Below is the graph of $\displaystyle f(x) = |x|$:

The given graph is the graph of $\displaystyle f$ reflected in the $\displaystyle x$-axis, then translated up 6 units. This graph is 

$\displaystyle g (x) = -f(x ) + k$, where $\displaystyle k = 6$.

The function graphed is therefore

$\displaystyle g (x) = -f(x ) + k$

$\displaystyle g (x) = -f(x ) + 6$

$\displaystyle g (x) = - |x|+ 6$

Below is the graph of $\displaystyle f(x) = |x|$:

The given graph is the graph of $\displaystyle f$ reflected in the $\displaystyle x$-axis, then translated left 2 units (or, equivalently, right $\displaystyle -2$ units. This graph is 

$\displaystyle g (x) = -f(x-h)$, where $\displaystyle h = -2$.

The function graphed is therefore

$\displaystyle g (x) = -f(x- (-2))$

$\displaystyle g (x) = -f(x+2)$

$\displaystyle g(x) = - |x +2|$

Below is the graph of $\displaystyle f(x) = |x|$:

The given graph is the graph of $\displaystyle f$  translated by moving the graph 7 units left (that is, $\displaystyle -7$ unit right) and 2 units down (that is, $\displaystyle -2$ units up)

The function graphed is therefore

$\displaystyle g (x) = f(x-h) + k$ where $\displaystyle h = -7, k= -2$. That is,

$\displaystyle g (x) = f(x-(-7)) + (-2)$

$\displaystyle g (x) = f(x+7) -2$

$\displaystyle g(x)= | x+7 | - 2$

The formula of an absolute value function is $\displaystyle y=|m(x-a)|+b$ where m is the slope, a is the horizontal shift and b is the vertical shift. The slope can be found with any two adjacent integer points, e.g. $\displaystyle (0,1)$ and $\displaystyle (1,3)$, and plugging them into the slope formula, $\displaystyle m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$, yielding $\displaystyle m=2$. The vertical and horizontal shifts are determined by where the crux of the absolute value function is. In this case, at $\displaystyle (-2,-3)$, and those are your a and b, respectively.

Let $\displaystyle g(x) = |x|$

The graph of this basic absolute value function is a "V"-shaped graph with a vertex at the origin, or the point with coordinates $\displaystyle (0,0)$. In terms of $\displaystyle g(x)$,

$\displaystyle f(x) = g(x + 6 )- 7$

The graph of this function can be formed by shifting the graph of $\displaystyle g(x)$ left 6 units ($\displaystyle +6$ ) and down 7 units ($\displaystyle -7$). The vertex is therefore located at $\displaystyle (-6, -7)$.

Let $\displaystyle g(x) = |x|$

The graph of this basic absolute value function is a "V"-shaped graph with a vertex at the origin, or the point with coordinates $\displaystyle (0,0)$. In terms of $\displaystyle g(x)$,

$\displaystyle f(x) = 10 - g (x+ 10)$, 

or, alternatively written,

$\displaystyle f(x) = - g (x+ 10)+ 10$

The graph of $\displaystyle f(x)$ is the same as that of $\displaystyle g(x)$, after it shifts 10 units left ( $\displaystyle g(x+10)$ ), it flips vertically (negative symbol), and it shifts up 10 units (the second $\displaystyle +10$). The flip does not affect the position of the vertex, but the shifts do; the vertex of the graph of $\displaystyle f(x)$ is at $\displaystyle (-10, 10)$.

The equation can be determined from the graph by following the rules of transformations; the base equation is:

$\displaystyle f(x)=\left | x\right |$

The graph of this base equation is:

When we compare our graph to the base equation graph, we see that it has been shifted right 3 units, up 1 unit, and our graph has been stretched vertically by a factor of 2. Following the rules of transformations, the equation for our graph is written as:

$\displaystyle f(x)=2\left | x-3\right |+1$