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Ontario Tech acknowledges the lands and people of the Mississaugas of Scugog Island First Nation.

We are thankful to be welcome on these lands in friendship. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. These lands remain home to many Indigenous nations and peoples.

We acknowledge this land out of respect for the Indigenous nations who have cared for Turtle Island, also called North America, from before the arrival of settler peoples until this day. Most importantly, we acknowledge that the history of these lands has been tainted by poor treatment and a lack of friendship with the First Nations who call them home.

This history is something we are all affected by because we are all treaty people in Canada. We all have a shared history to reflect on, and each of us is affected by this history in different ways. Our past defines our present, but if we move forward as friends and allies, then it does not have to define our future.

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Solving Linear Inequalities

An inequality is a statement indicating that two expressions are not equal to one another in a particular way (e.g. one expression is larger or smaller than the other). In the case of a linear inequality, it can be simplified into the form

ax ≥ b, ax b, ax ≤ b, or ax b

where a and b are real numbers and a ≠ 0. 

Solving a Linear Inequality

To solve a linear inequality, you have to isolate for the variable by doing the following steps:

  • Expand (if applicable)
  • Group like terms (if applicable)
  • Rearrange so that all terms with the variable in them are on one side of the inequality while all the terms without the variable in them (i.e. just number terms) are on the other side. That is, rearrange into the form ax > b or ax b, etc. (to do this, you're just adding/subtracting terms from both sides)
  • Divide by the coefficient of the variable to solve for the variable (i.e., once you've got ax > b or ax b, etc., divide both sides by a to solve for x). IMPORTANT: if you divide by a negative number, the inequality switches direction (i.e., > becomes < and so on)

Example: Solve the linear inequality 5x - 1  3x + 2


5x - 1  3x + 2

5x - 3x  2+ 1

        2x  3

         x  3/2

Example: Solve the inequality -6x + 1 < 3x + 4


-6x +1 < 3x + 4

-6x - 3x < 4 - 1

       -9x < 3

          x > -1/3

Note that we could have avoided the situation having to divide by a negative number (and hence changing the sign of the inequality) by collecting the terms with the variable on the right of the inequality and the terms with only numbers on the left. Let's do the question this way as well so that you can see the difference:

-6x +1 < 3x + 4

    1 - 4 < 3x + 6x

        -3 < 9x

     -3/9 x

     -1/3 x

Of course, even though it might not look like it at first glance, -1/3 < x and x -1/3 are saying the exact same thing. What we're saying in both cases is: x is greater than - 1/3

Linear Example 1:

Linear Example 2:

Quadratic Example:

Three Factors Example:

Absolute Value Inequalities Example 1:

Absolute Value Inequalities Example 2: