What is a linear inequality? A linear inequality is a linear function. There are symbols involved in this, < is less than. > is greater than. ≤ is less than or equal to.

How do you solve them? If it is a multi-step equation, then solve it as you normally would and remove the zero pairs.  At the end, you have to divide to find what x is. If you have a negative number, you have to flip the symbol around, as seen in the bottom of this solved equation.

After finishing the equation, you have to graph the answer in order to understand it better. If you have a closed dot on the number line, it means that the number is either more than, less than, or equal to x. If you have an open dot on the number line, it means that the number is either less than or greater than x. How do you place it on the number lines? Well, greater than means that you go to the right, while less than means that you go to the left.

This graph here would be 4.2>x.

What is a linear equation? A linear equation is an equation between 2 variables that when plotted on a graph, gives a straight line. It has an equals sign in between the 2 halves of the equation to join it together. The 2 variables can be in fraction form, decimal, integer or whole number. If you are using fractions, the easiest way to simplify the equation is to find a common denominator, which Is a number that all the denominators have in comon. For example, ‘1/3x+2/6=8/12x+1/3’, would have a CD of 12. Changing the bottom denominator means you also have to change the top number, and multiply it by whatever you multiplied the denominator by. ‘4/12x+4/12=8/12x+4/12’.

Linear equations can also be equivalent expressions. These are algebraic expressions that are made equal by changing the values of the numbers. An equivalent expression of 4x+3=-6x-2 would be 4x+3=2(3x+1) and 10x+5=0.

How can equations be modelled? They can be modelled using algebra tiles in paper form or drawn on. The dark tiles represent positive integers, and the white or non-coloured tiles represent negative integers. The dark rectangular tiles are used for 1x, and the smaller cubes are used for 1.  The white rectangular tiles are used for -1x and the smaller cubes are used for -1.

Equations can be solved visually many ways. One way is to use algebra tiles, and another way is to draw the tiles onto the paper on the correct formatting of the equation.

One way you can solve equations algebraically is by grouping the like terms together and solving it from there.  The distributive method is when you have a whole number and you have to distribute it to numbers inside the brackets.  An easy way to verify the solution is to work the question backwards and to pretend like you don’t know what x is. You can also find the zero pairs if you have ‘3x-2=4x+3’, the first zero pair would be ‘-3x’, which cancels out ‘+3x’. The equation would then be turned into ‘-2=1x+3’ and the next zero pair would be ‘-3’. Thus, the simplified equation would be ‘-5=1x’.

In the expression ‘4x-7=5’, 4 would be the coefficient, since it is the value that comes before the variable, which is x.  The constant term would be -7 and =5, since those are the things that do not change. The operator is subtraction. A solution is having the ability to check that no matter what you are using as x, 4x-7 will always equal to 5.

Here, for this equation below, I first started by making zero pairs with the negative 4x by adding positive 4x. After that, I added 8 to make zero pairs with the 8. After that was finished, I ended up with ‘+10=10x’, and if I divide 10 by 10x, then I get 1x=1. I also solved It with algebra tiles.

The picture above is me solving an equation using the regular strategy of how I normally solve it. The picture below is me explaining how to solve it with algebra tiles.

What is a Linear Relation? The words “Linear Relation “, are used to describe the relationship between the input and output of data. This data can be shown through many ways.

One way is a graph, the x-axis runs horizontally, while the y-axis runs vertically. In order to properly place a point on the graph, you must find where the positives and negatives are placed. On the left x-axis, the numbers are negative, on the bottom of the y-axis, the numbers are also negative. On the right side of the x-axis, the numbers are positive, the same as the top half of the y-axis.

You can also plot this in a t-chart. On the left side of the chart, the equivalent of the x-axis,  will be the input or step number. As you move to the right side, the same as the y-axis,  you get the output, or the result. In order to make a rule, you must find the relations between the two forms of data.

There are 2 types of patterns. One is a decreasing pattern, which is when you get a larger number for the input then you do for the output. Therefore, the line starts in the top left, and gradually goes down diagonally to the bottom right. The values and data in this type gradually get smaller, decreasing.

The other type of pattern is increasing. The pattern gradually or suddenly gets larger and larger each time. This pattern below is increasing, as the line starts in the bottom left, and gets larger, and moves to the top right.

If you plot this data on a graph, you will get either a straight or diagonal line. As shown in this graph below, the line is increasing on a horizontal angle.