Every matrix has a dimension, which is represented as the number of rows and columns. For example, a matrix with three rows and two columns is said to have dimension 3 x 2. 

The matrix 

\(\displaystyle \begin{bmatrix} -1&2 &0 \\ 4&-3 & 0 \end{bmatrix}\)

has two rows and three columns, so its dimension is 2 x 3. (Remember that rows go from left to right, while columns run up and down.)

Matrix multiplication is defined only if the number of columns in the first matrix is equal to the number of rows on the second matrix. The easiest way to determine this is to write the dimension of each matrix. For example, let's say that one matrix has dimension a x b, and the second matrix has dimension c x d. We can only multiply the first matrix by the second matrix if the values of b and c are equal. It doesn't matter what the values of a and d are, as long as b (the number of columns in the first matrix) matches c (the number of rows in the second matrix).

Let's go back to the problem and analyze the choice \(\displaystyle \begin{bmatrix} -1&2 &0 \\ 4&-3 & 0 \end{bmatrix}\cdot\begin{bmatrix} -1 & 2 \\ 4& -3& \end{bmatrix}\).

The dimension of the first matrix is 2 x 3, because it has two rows and three columns. The second matrix has dimension 2 x 2, because it has two rows and two columns.

We can't multiply these matrices because the number of columns in the first matrix (3) is not equal to the number of rows in the second matrix (2). Thus, this product is not defined.

The answer is \(\displaystyle \begin{bmatrix} -1&2 &0 \\ 4&-3 & 0 \end{bmatrix}\cdot\begin{bmatrix} -1 & 2 \\ 4& -3& \end{bmatrix}\).

Convert the mixed numbers into improper fractions by multiplying the whole number by the denominator and adding the numerator to get \(\displaystyle \frac{13}{3}\div \frac{5}{2}\)

Dividing by a fraction is the same as multiplying by its reciprocal so the problem becomes \(\displaystyle \frac{13}{3}\cdot \frac{2}{5} = \frac{26}{15} =1\frac{11}{15}\)

How do you find the area of a hexagon?

There are several ways to find the area of a hexagon.

1. In a regular hexagon, split the figure into triangles.
2. Find the area of one triangle.
3. Multiply this value by six.

Alternatively, the area can be found by calculating one-half of the side length times the apothem.

Regular hexagons:

Regular hexagons are interesting polygons. Hexagons are six sided figures and possess the following shape:

In a regular hexagon, all sides equal the same length and all interior angles have the same measure; therefore, we can write the following expression.

\(\displaystyle \overline{AB}=\overline{BC}=\overline{CD}=\overline{DE}=\overline{EF}=\overline{FA}\) 

One of the easiest methods that can be used to find the area of a polygon is to split the figure into triangles. Let's start by splitting the hexagon into six triangles. 

In this figure, the center point, \(\displaystyle X\), is equidistant from all of the vertices. As a result, the six dotted lines within the hexagon are the same length. Likewise, all of the triangles within the hexagon are congruent by the side-side-side rule: each of the triangle's share two sides inside the hexagon as well as a base side that makes up the perimeter of the hexagon. In a similar fashion, each of the triangles have the same angles. There are \(\displaystyle 360^{\circ}\) in a circle and the hexagon in our image has separated it into six equal parts; therefore, we can write the following:

\(\displaystyle \theta=\frac{360^{\circ}}{6}\)

\(\displaystyle \theta=60^{\circ\)

We also know the following:

\(\displaystyle \angle AXB=\angle BXC=\angle CXD=\angle DXE=\angle EXF=\angle FXA=60^{\circ}\)

Now, let's look at each of the triangles in the hexagon. We know that each triangle has two two sides that are equal; therefore, each of the base angles of each triangle must be the same. We know that a triangle has \(\displaystyle 180^{\circ}\) and we can solve for the two base angles of each triangle using this information.

\(\displaystyle 2x+60^{\circ}=180^{\circ}\)

\(\displaystyle 2x+=120^{\circ}\)

\(\displaystyle x=60^{\circ}\)

Each angle in the triangle equals \(\displaystyle 60^{\circ}\). We now know that all the triangles are congruent and equilateral: each triangle has three equal side lengths and three equal angles. Now, we can use this vital information to solve for the hexagon's area. If we find the area of one of the triangles, then we can multiply it by six in order to calculate the area of the entire figure. Let's start by analyzing \(\displaystyle \triangle BXC\). If we draw, an altitude through the triangle, then we find that we create two \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangles. 

Let's solve for the length of this triangle. Remember that in \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangles, triangles possess side lengths in the following ratio:

\(\displaystyle 1:2:\sqrt{3}\)

Now, we can analyze \(\displaystyle \triangle BXC\) using the a substitute variable for side length, \(\displaystyle s\).

We know the measure of both the base and height of \(\displaystyle \triangle BXC\) and we can solve for its area.

\(\displaystyle \textup{Area of }\triangle BXC=\frac{1}{2}\times \textup{base}\times \textup{height}\)

\(\displaystyle \textup{Area of }\triangle BXC=\frac{1}{2}\times s\times \frac{\sqrt{3}s}{2}\)

\(\displaystyle \textup{Area of }\triangle BXC=\frac{\sqrt{3}s^2}{4}\)

Now, we need to multiply this by six in order to find the area of the entire hexagon.

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=6\times\frac{\sqrt{3}s^2}{4}\)

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=\frac{6 \sqrt{3}s^2}{4}\)

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=\frac{3\sqrt{3}}{2}\times s^2\)

We have solved for the area of a regular hexagon with side length, \(\displaystyle s\). If we know the side length of a regular hexagon, then we can solve for the area.

If we are not given a regular hexagon, then we an solve for the area of the hexagon by using the side length(i.e. \(\displaystyle s\)) and apothem (i.e. \(\displaystyle a\)), which is the length of a line drawn from the center of the polygon to the right angle of any side. This is denoted by the variable \(\displaystyle a\) in the following figure:

Alternative method:

If we are given the variables \(\displaystyle s\) and \(\displaystyle a\), then we can solve for the area of the hexagon through the following formula:

\(\displaystyle A=\frac{1}{2}(P)(a)\)

In this equation, \(\displaystyle A\) is the area, \(\displaystyle P\) is the perimeter, and \(\displaystyle a\) is the apothem. We must calculate the perimeter using the side length and the equation \(\displaystyle P=\textup{number of sides}\times s\), where \(\displaystyle s\) is the side length.

Solution:

In the given problem we know that the side length of a regular hexagon is the following:

\(\displaystyle s=6\)

Let's substitute this value into the area formula for a regular hexagon and solve.

\(\displaystyle A=\frac{3\sqrt{3}}{2}\times s^2\)

\(\displaystyle A=\frac{3\sqrt{3}}{2}\times 6^2\)

Simplify.

\(\displaystyle A=\frac{108\sqrt{3}}{2}\)

\(\displaystyle A=54\sqrt{3}\)

\(\displaystyle A=93.53\)

Round the answer to the nearest whole number.

\(\displaystyle A=94\)

How do you find the area of a hexagon?

There are several ways to find the area of a hexagon.

1. In a regular hexagon, split the figure into triangles.
2. Find the area of one triangle.
3. Multiply this value by six.

Alternatively, the area can be found by calculating one-half of the side length times the apothem.

Regular hexagons:

Regular hexagons are interesting polygons. Hexagons are six sided figures and possess the following shape:

In a regular hexagon, all sides equal the same length and all interior angles have the same measure; therefore, we can write the following expression.

\(\displaystyle \overline{AB}=\overline{BC}=\overline{CD}=\overline{DE}=\overline{EF}=\overline{FA}\) 

One of the easiest methods that can be used to find the area of a polygon is to split the figure into triangles. Let's start by splitting the hexagon into six triangles. 

In this figure, the center point, \(\displaystyle X\), is equidistant from all of the vertices. As a result, the six dotted lines within the hexagon are the same length. Likewise, all of the triangles within the hexagon are congruent by the side-side-side rule: each of the triangle's share two sides inside the hexagon as well as a base side that makes up the perimeter of the hexagon. In a similar fashion, each of the triangles have the same angles. There are \(\displaystyle 360^{\circ}\) in a circle and the hexagon in our image has separated it into six equal parts; therefore, we can write the following:

\(\displaystyle \theta=\frac{360^{\circ}}{6}\)

\(\displaystyle \theta=60^{\circ\)

We also know the following:

\(\displaystyle \angle AXB=\angle BXC=\angle CXD=\angle DXE=\angle EXF=\angle FXA=60^{\circ}\)

Now, let's look at each of the triangles in the hexagon. We know that each triangle has two two sides that are equal; therefore, each of the base angles of each triangle must be the same. We know that a triangle has \(\displaystyle 180^{\circ}\) and we can solve for the two base angles of each triangle using this information.

\(\displaystyle 2x+60^{\circ}=180^{\circ}\)

\(\displaystyle 2x+=120^{\circ}\)

\(\displaystyle x=60^{\circ}\)

Each angle in the triangle equals \(\displaystyle 60^{\circ}\). We now know that all the triangles are congruent and equilateral: each triangle has three equal side lengths and three equal angles. Now, we can use this vital information to solve for the hexagon's area. If we find the area of one of the triangles, then we can multiply it by six in order to calculate the area of the entire figure. Let's start by analyzing \(\displaystyle \triangle BXC\). If we draw, an altitude through the triangle, then we find that we create two \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangles. 

Let's solve for the length of this triangle. Remember that in \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangles, triangles possess side lengths in the following ratio:

\(\displaystyle 1:2:\sqrt{3}\)

Now, we can analyze \(\displaystyle \triangle BXC\) using the a substitute variable for side length, \(\displaystyle s\).

We know the measure of both the base and height of \(\displaystyle \triangle BXC\) and we can solve for its area.

\(\displaystyle \textup{Area of }\triangle BXC=\frac{1}{2}\times \textup{base}\times \textup{height}\)

\(\displaystyle \textup{Area of }\triangle BXC=\frac{1}{2}\times s\times \frac{\sqrt{3}s}{2}\)

\(\displaystyle \textup{Area of }\triangle BXC=\frac{\sqrt{3}s^2}{4}\)

Now, we need to multiply this by six in order to find the area of the entire hexagon.

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=6\times\frac{\sqrt{3}s^2}{4}\)

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=\frac{6 \sqrt{3}s^2}{4}\)

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=\frac{3\sqrt{3}}{2}\times s^2\)

We have solved for the area of a regular hexagon with side length, \(\displaystyle s\). If we know the side length of a regular hexagon, then we can solve for the area.

If we are not given a regular hexagon, then we an solve for the area of the hexagon by using the side length(i.e. \(\displaystyle s\)) and apothem (i.e. \(\displaystyle a\)), which is the length of a line drawn from the center of the polygon to the right angle of any side. This is denoted by the variable \(\displaystyle a\) in the following figure:

Alternative method:

If we are given the variables \(\displaystyle s\) and \(\displaystyle a\), then we can solve for the area of the hexagon through the following formula:

\(\displaystyle A=\frac{1}{2}(P)(a)\)

In this equation, \(\displaystyle A\) is the area, \(\displaystyle P\) is the perimeter, and \(\displaystyle a\) is the apothem. We must calculate the perimeter using the side length and the equation \(\displaystyle P=\textup{number of sides}\times s\), where \(\displaystyle s\) is the side length.

Solution:

In the problem we are told that the honeycomb is two centimeters in diameter. In order to solve the problem we need to divide the diameter by two. This is because the radius of this diameter equals the interior side length of the equilateral triangles in the honeycomb. Lets find the side length of the regular hexagon/honeycomb.

\(\displaystyle \textup{Diameter}=2\times \textup{ radius}\)

Substitute and solve.

\(\displaystyle 2\textup{ cm}=2\times \textup{radius}\)

\(\displaystyle \textup{radius}=\frac{2\textup{ cm}}{2}\)

\(\displaystyle \textup{radius}=1\textup{ cm}\)

We know the following information.

\(\displaystyle \textup{radius}=\textup{side length}\)

As a result, we can write the following:

\(\displaystyle s=1\textup{ cm}\)

Let's substitute this value into the area formula for a regular hexagon and solve.

\(\displaystyle A=\frac{3\sqrt{3}}{2}\times s^2\)

\(\displaystyle A=\frac{3\sqrt{3}}{2}\times (1\textup{ cm})^2\)

Simplify.

\(\displaystyle A=\frac{3\sqrt{3}}{2} \textup{ cm}^2\)

Solve.

\(\displaystyle A=2.59\textup{ cm}^2\)

Round to the nearest tenth of a centimeter.

\(\displaystyle A=2.6\textup{ cm}^2\)

How do you find the area of a hexagon?

There are several ways to find the area of a hexagon.

1. In a regular hexagon, split the figure into triangles.
2. Find the area of one triangle.
3. Multiply this value by six.

Alternatively, the area can be found by calculating one-half of the side length times the apothem.

Regular hexagons:

Regular hexagons are interesting polygons. Hexagons are six sided figures and possess the following shape:

In a regular hexagon, all sides equal the same length and all interior angles have the same measure; therefore, we can write the following expression.

\(\displaystyle \overline{AB}=\overline{BC}=\overline{CD}=\overline{DE}=\overline{EF}=\overline{FA}\) 

One of the easiest methods that can be used to find the area of a polygon is to split the figure into triangles. Let's start by splitting the hexagon into six triangles. 

In this figure, the center point, \(\displaystyle X\), is equidistant from all of the vertices. As a result, the six dotted lines within the hexagon are the same length. Likewise, all of the triangles within the hexagon are congruent by the side-side-side rule: each of the triangle's share two sides inside the hexagon as well as a base side that makes up the perimeter of the hexagon. In a similar fashion, each of the triangles have the same angles. There are \(\displaystyle 360^{\circ}\) in a circle and the hexagon in our image has separated it into six equal parts; therefore, we can write the following:

\(\displaystyle \theta=\frac{360^{\circ}}{6}\)

\(\displaystyle \theta=60^{\circ\)

We also know the following:

\(\displaystyle \angle AXB=\angle BXC=\angle CXD=\angle DXE=\angle EXF=\angle FXA=60^{\circ}\)

Now, let's look at each of the triangles in the hexagon. We know that each triangle has two two sides that are equal; therefore, each of the base angles of each triangle must be the same. We know that a triangle has \(\displaystyle 180^{\circ}\) and we can solve for the two base angles of each triangle using this information.

\(\displaystyle 2x+60^{\circ}=180^{\circ}\)

\(\displaystyle 2x+=120^{\circ}\)

\(\displaystyle x=60^{\circ}\)

Each angle in the triangle equals \(\displaystyle 60^{\circ}\). We now know that all the triangles are congruent and equilateral: each triangle has three equal side lengths and three equal angles. Now, we can use this vital information to solve for the hexagon's area. If we find the area of one of the triangles, then we can multiply it by six in order to calculate the area of the entire figure. Let's start by analyzing \(\displaystyle \triangle BXC\). If we draw, an altitude through the triangle, then we find that we create two \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangles. 

Let's solve for the length of this triangle. Remember that in \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangles, triangles possess side lengths in the following ratio:

\(\displaystyle 1:2:\sqrt{3}\)

Now, we can analyze \(\displaystyle \triangle BXC\) using the a substitute variable for side length, \(\displaystyle s\).

We know the measure of both the base and height of \(\displaystyle \triangle BXC\) and we can solve for its area.

\(\displaystyle \textup{Area of }\triangle BXC=\frac{1}{2}\times \textup{base}\times \textup{height}\)

\(\displaystyle \textup{Area of }\triangle BXC=\frac{1}{2}\times s\times \frac{\sqrt{3}s}{2}\)

\(\displaystyle \textup{Area of }\triangle BXC=\frac{\sqrt{3}s^2}{4}\)

Now, we need to multiply this by six in order to find the area of the entire hexagon.

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=6\times\frac{\sqrt{3}s^2}{4}\)

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=\frac{6 \sqrt{3}s^2}{4}\)

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=\frac{3\sqrt{3}}{2}\times s^2\)

We have solved for the area of a regular hexagon with side length, \(\displaystyle s\). If we know the side length of a regular hexagon, then we can solve for the area.

If we are not given a regular hexagon, then we an solve for the area of the hexagon by using the side length(i.e. \(\displaystyle s\)) and apothem (i.e. \(\displaystyle a\)), which is the length of a line drawn from the center of the polygon to the right angle of any side. This is denoted by the variable \(\displaystyle a\) in the following figure:

Alternative method:

If we are given the variables \(\displaystyle s\) and \(\displaystyle a\), then we can solve for the area of the hexagon through the following formula:

\(\displaystyle A=\frac{1}{2}(P)(a)\)

In this equation, \(\displaystyle A\) is the area, \(\displaystyle P\) is the perimeter, and \(\displaystyle a\) is the apothem. We must calculate the perimeter using the side length and the equation \(\displaystyle P=\textup{number of sides}\times s\), where \(\displaystyle s\) is the side length.

Solution:

In a hexagon the number of sides is \(\displaystyle 6\) and in this example the side length is \(\displaystyle 7\).

\(\displaystyle P=6\times7=42\)

The perimeter is \(\displaystyle 42\).

Then we plug in the numbers for the apothem and perimeter into the original equation.

\(\displaystyle A=\frac{1}{2}(P)(a)=\frac{1}{2}(42)(4)=(21)(4)=84\)

The area is \(\displaystyle 84\).

How do you find the area of a hexagon?

There are several ways to find the area of a hexagon.

1. In a regular hexagon, split the figure into triangles.
2. Find the area of one triangle.
3. Multiply this value by six.

Alternatively, the area can be found by calculating one-half of the side length times the apothem.

Regular hexagons:

Regular hexagons are interesting polygons. Hexagons are six sided figures and possess the following shape:

In a regular hexagon, all sides equal the same length and all interior angles have the same measure; therefore, we can write the following expression.

\(\displaystyle \overline{AB}=\overline{BC}=\overline{CD}=\overline{DE}=\overline{EF}=\overline{FA}\) 

One of the easiest methods that can be used to find the area of a polygon is to split the figure into triangles. Let's start by splitting the hexagon into six triangles. 

In this figure, the center point, \(\displaystyle X\), is equidistant from all of the vertices. As a result, the six dotted lines within the hexagon are the same length. Likewise, all of the triangles within the hexagon are congruent by the side-side-side rule: each of the triangle's share two sides inside the hexagon as well as a base side that makes up the perimeter of the hexagon. In a similar fashion, each of the triangles have the same angles. There are \(\displaystyle 360^{\circ}\) in a circle and the hexagon in our image has separated it into six equal parts; therefore, we can write the following:

\(\displaystyle \theta=\frac{360^{\circ}}{6}\)

\(\displaystyle \theta=60^{\circ\)

We also know the following:

\(\displaystyle \angle AXB=\angle BXC=\angle CXD=\angle DXE=\angle EXF=\angle FXA=60^{\circ}\)

Now, let's look at each of the triangles in the hexagon. We know that each triangle has two two sides that are equal; therefore, each of the base angles of each triangle must be the same. We know that a triangle has \(\displaystyle 180^{\circ}\) and we can solve for the two base angles of each triangle using this information.

\(\displaystyle 2x+60^{\circ}=180^{\circ}\)

\(\displaystyle 2x+=120^{\circ}\)

\(\displaystyle x=60^{\circ}\)

Each angle in the triangle equals \(\displaystyle 60^{\circ}\). We now know that all the triangles are congruent and equilateral: each triangle has three equal side lengths and three equal angles. Now, we can use this vital information to solve for the hexagon's area. If we find the area of one of the triangles, then we can multiply it by six in order to calculate the area of the entire figure. Let's start by analyzing \(\displaystyle \triangle BXC\). If we draw, an altitude through the triangle, then we find that we create two \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangles. 

Let's solve for the length of this triangle. Remember that in \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangles, triangles possess side lengths in the following ratio:

\(\displaystyle 1:2:\sqrt{3}\)

Now, we can analyze \(\displaystyle \triangle BXC\) using the a substitute variable for side length, \(\displaystyle s\).

We know the measure of both the base and height of \(\displaystyle \triangle BXC\) and we can solve for its area.

\(\displaystyle \textup{Area of }\triangle BXC=\frac{1}{2}\times \textup{base}\times \textup{height}\)

\(\displaystyle \textup{Area of }\triangle BXC=\frac{1}{2}\times s\times \frac{\sqrt{3}s}{2}\)

\(\displaystyle \textup{Area of }\triangle BXC=\frac{\sqrt{3}s^2}{4}\)

Now, we need to multiply this by six in order to find the area of the entire hexagon.

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=6\times\frac{\sqrt{3}s^2}{4}\)

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=\frac{6 \sqrt{3}s^2}{4}\)

\(\displaystyle \textup{Area of Hexagon}[ABCDEF]=\frac{3\sqrt{3}}{2}\times s^2\)

We have solved for the area of a regular hexagon with side length, \(\displaystyle s\). If we know the side length of a regular hexagon, then we can solve for the area.

If we are not given a regular hexagon, then we an solve for the area of the hexagon by using the side length(i.e. \(\displaystyle s\)) and apothem (i.e. \(\displaystyle a\)), which is the length of a line drawn from the center of the polygon to the right angle of any side. This is denoted by the variable \(\displaystyle a\) in the following figure:

Alternative method:

If we are given the variables \(\displaystyle s\) and \(\displaystyle a\), then we can solve for the area of the hexagon through the following formula:

\(\displaystyle A=\frac{1}{2}(P)(a)\)

In this equation, \(\displaystyle A\) is the area, \(\displaystyle P\) is the perimeter, and \(\displaystyle a\) is the apothem. We must calculate the perimeter using the side length and the equation \(\displaystyle P=\textup{number of sides}\times s\), where \(\displaystyle s\) is the side length.

Solution:

In the given problem we know that the side length of a regular hexagon is the following:

\(\displaystyle s=6\textup{ cm}\)

Let's substitute this value into the area formula for a regular hexagon and solve.

\(\displaystyle A=\frac{3\sqrt{3}}{2}\times s^2\)

\(\displaystyle A=\frac{3\sqrt{3}}{2}\times (6\textup{ cm})^2\)

Simplify.

\(\displaystyle A=\frac{108\sqrt{3}}{2} \textup{ cm}^{2}\)

\(\displaystyle A=54\sqrt{3} \textup{ cm}^{2}\)

\(\displaystyle A=93.53 \textup{ cm}^{2}\)

In finding the values for \(\displaystyle x\) where \(\displaystyle \left | x -3 \right | > \left |x+12 \right |\), break the comparison of these two absolute value expressions into the four possible ways this could potentially be satisfied.

The first possibility is described by the inequality:

\(\displaystyle x-3 > x+12\)

If you think of a number line, it is evident that there is no solution to this inequality since there will never be a case where subtracting from \(\displaystyle x\) will lead to a greater number than adding to \(\displaystyle x\).

The second possibility, wherein \(\displaystyle x-3\) is negative and converted to its opposite to being an absolute value expression but \(\displaystyle x+12\) is positive and requires no conversion, can be represented by the inequality (where the sign is inverted due to multiplication by a negative):

\(\displaystyle -(x-3) < x+12\)

We can simplify this inequality to find that \(\displaystyle x< -4.5\) satisfies the conditions where \(\displaystyle \left | x -3 \right | > \left |x+12 \right |\).

The third possibility can be represented by the following inequality (where the sign is inverted due to multiplication by a negative):

\(\displaystyle x-3 < -(x+12)\)

This is again simplified to \(\displaystyle x< -4.5\) and is redundant with the above inequality.

The final possibility is represented by the inequality

\(\displaystyle -(x-3) > -(x+12)\)

This inequality simplifies to \(\displaystyle 3 - x > -12 - x\). Rewriting this as \(\displaystyle - x + 3 > - x -12\) makes it evident that this inequality is true of all real numbers. This does not provide any additional conditions on how to satisfy the original inequality.

The only possible condition that satisfies the inequality is that which arises in two of the tested cases, when \(\displaystyle x < -4.5\).

The absolute value is the distance from a given number to 0. In our example, we are given -3. This number is 3 units away from 0, and thus the absolute value of -3 is 3. 

If a number is negative, its absolute value will be the positive number with the same magnitude. If a number is positive, it will be its own absolute value. 

Notice that the equation \(\displaystyle 2x + \left | x - 3 \right | = -12\) has an \(\displaystyle x\) term both inside and outside the absolute value expression.

Since the absolute value expression will always produce a positive number and the right side of the equation is negative, a negative number must be added to the result of the absolute value expression to satisfy the equation. Therefore the term outside of the absolute value expression (in this case \(\displaystyle 2x\)) must be negative (meaning \(\displaystyle x\) must be negative). 

Since \(\displaystyle x\) will be a negative number, the expression within the absolute value \(\displaystyle x-3\) will also be negative (before the absolute value is taken). It is thus possible to convert the original equation into an equation that treats the absolute value as a parenthetical expression that will be multiplied by \(\displaystyle -1\), since any negative value becomes its opposite when taking the absolute value.

\(\displaystyle 2x + \left | x - 3 \right | = -12\)

\(\displaystyle \rightarrow 2x+ (x-3)(-1) = -12\)

Simplifying and solving this equation for \(\displaystyle x\) gives the answer:

\(\displaystyle \rightarrow 2x -x+3 = -12\)

\(\displaystyle \rightarrow x + 3 = -12\)

\(\displaystyle \rightarrow x = -15\)

The absolute value measures the distance from zero to the given point. 

In this case, since \(\displaystyle |x|=12\), \(\displaystyle x=12\) or \(\displaystyle x=-12\), as both values are twelve units away from zero.