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A short class with Sarah and her students. The other ways are under More.
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Today, we’re discussing linear inequalities in one variable. Can anyone tell me how they differ from regular equations?
Linear inequalities use symbols like < and > instead of just =.
Exactly! While equations show equality between two expressions, inequalities compare them, resulting in a range of values. Now, let's look at how to solve these inequalities. What do we need to remember?
We have to reverse the sign when we multiply or divide by a negative number!
Great memory! Remember, we use the acronym 'RIF'—Reverse If Negative—to help us remember. Let's try a sample inequality together.
Like 2x - 3 < 5?
Exactly! How do we tackle that one?
First, we add 3 to both sides, giving us 2x < 8.
Perfect! And what’s the next step?
We divide by 2, so x < 4!
Well done! Let’s summarize: Linear inequalities use different symbols and require special rules when we multiply or divide by negatives. Let’s move on to graphing these solutions.
Now, let’s talk about graphing our solutions. Can someone remind me how we represent the solutions on a number line?
We use open circles for < or > and closed circles for ≤ or ≥.
Excellent! And what does shading indicate?
It shows the range of solutions that satisfy the inequality!
Correct! Let’s visualize this using the example we solved earlier, x < 4. How would that look?
We place an open circle on 4 and shade everything to the left!
Exactly! This gives us a complete visual representation of the solutions. Always remember to check the direction of the inequality as you shade.
So, the open circle means 4 isn’t included?
Yes! Now let’s move to the next session and explore examples with inequalities containing negative coefficients.
Now, let’s discuss solving inequalities that involve negatives. For instance, how do we solve -3x + 4 ≥ 10?
We start by subtracting 4 from both sides to get -3x ≥ 6.
Correct! And what happens next?
We divide by -3, so we must reverse the inequality, leading to x ≤ -2.
Exactly! This important step is crucial whenever you deal with negatives. Can someone summarize our key points from today?
Always remember to reverse the sign when dividing by a negative and graph with open or closed circles depending on the symbol!
Great summary! Now you have an excellent foundation for solving and graphing linear inequalities. Let's wrap up with some practical exercises.
Use symbols to represent ranges of possible solutions.
Necessary when multiplying or dividing by negative numbers.
Open and closed circles represent inclusion and exclusion.
Indicates the range of values that satisfy the inequality.
When a negative's in the game, flip the sign to keep the same!
Imagine a seesaw: when you push down (negative), the other side must lift up (reverse the sign) to balance it out.
RIF - Remember If Negative: Flip the sign!
GLOW - Graphing Line with Open or closed circle, depending on the sign.
Term
What happens when dividing by a negative in inequalities?
Click to flip
Term
How is x < 4 represented on a number line?
Click to flip
Term
What do we call the range of solutions in inequalities?
Click to flip
An inequality that involves a linear expression and uses symbols such as <, ≤, >, or ≥.
The range of values that satisfy a given inequality.
A symbol used on a number line to indicate that a point is not included in the solution set (used for < or >).
A symbol used on a number line to indicate that a point is included in the solution set (used for ≤ or ≥).
The necessity to reverse the inequality sign when multiplying or dividing both sides of an inequality by a negative number.
This section explains how to solve linear inequalities in one variable, highlighting key concepts such as reversing inequality signs during multiplication or division by negative numbers and graphing solutions on a number line.
In this section, we will explore linear inequalities in one variable, covering essential rules for solving them, such as the need to reverse the inequality sign when multiplying or dividing by a negative number. We will also discuss how to represent solutions graphically on a number line using open and closed circles.
This section focuses on solving linear inequalities in one variable, which operates under similar principles as linear equations but requires additional consideration in certain operations. Here are the key points discussed:
2x - 3 < 5, the solution leads to x < 4.-3x + 4 ≥ 10, after correctly flipping the inequality sign, the result is x ≤ -2.This foundational understanding of linear inequalities sets the stage for grappling with more complex concepts later in the chapter, including inequalities in two variables.