Frequently asked questions about equations and inequalities
What are equations and why do we need them?
Equations are like math sentences that use an equal sign (=) to show that two expressions have the same value. For example,
is an equation that says adding and gives the same result as adding and .
We can also use letters, called variables, to represent unknown numbers in equations. For example,
is an equation that says adding to some number gives the same result as .
We can use equations to model real-world situations, such as how much money we have, how fast we are going, or how many crops we can grow. Equations are a useful model whenever we have two mathematical ways of naming the same quantity.
How do we test solutions to equations?
To test if a number is a solution to an equation, we can plug it in for the variable and see if the equation is true.
For example, to test if
is a solution to , we can replace with and get . This is true, so is a solution.
To test if
is a solution, we can replace with and get . This is false, so is not a solution.
What are one-step equations and how do we solve them?
One-step equations are equations that only have one operation, such as addition, subtraction, multiplication, or division. For example,
and are one-step equations.
To solve a one-step equation, we need to do the opposite operation to both sides of the equation to isolate the variable. For example, to solve
, we need to subtract from both sides to get .
, we need to multiply by both sides to get . Solving one-step equations helps us find the value of the unknown number that makes the equation true.
How do we test solutions to basic inequalities?
An inequality often has a whole set of solutions, of numbers that makes the statement true. Let's think about it. If we had the statement
, what are some values of that would make the statement true? There are lots of numbers less than , including many fractions and negative numbers.
To test if a number is a solution to an inequality, we can plug it in for the variable and see if the inequality is true. For example, to test if
is a solution to , we can replace with and get . This is true, so is a solution. To test if is a solution, we can replace with and get . This is false, so is not a solution to the inequality.
How do we graph basic inequalities on a number line?
In a basic inequality, we have just the variable on one side of the inequality, and just a number on the other side. If the inequality is not in that form yet, we will need to rewrite it to that form before we can graph it.
To graph basic inequalities on a number line, we need to mark the point that represents the boundary of the inequality, such as
for . Then, we need to use a circle to show if the boundary is included or excluded in the solution. If the inequality has an equal sign, such as or , we use a closed circle (●) to show that the boundary is included. If the inequality does not have an equal sign, such as or , we use an open circle (○) to show that the boundary is excluded.
Then, we need to shade the part of the number line that represents the solutions to the inequality. Pick a value that is either to the left or right of the boundary, such as
. Substitute that value into the inequality and see whether it makes the statement true. If so, shade that side of the number line. If not, try a number on the other side of the boundary. For example, to graph , we use an open circle at . The statement is false, so we would not shade the side of the number line with in it, which is to the left of the boundary. However, is true, so we would shade the right side of the number line, the side with the in it. To graph , we use a closed circle at and shade to the left.
Misconception alert: Some folks automatically shade to the right of the boundary when they see a greater than symbol like
or , but this only works if the isolated variable is to the left of the symbol. In the inequality , we would shade to the left, because all of the solutions are less than .
What are dependent and independent variables and how do we identify them?
Sometimes, when we do experiments or compare things, we want to know how one thing affects another. For example, we might want to know how the height of a ramp affects the speed of a toy car, or how the temperature of water affects the amount of sugar that can dissolve in it. In these situations, we use two types of variables: independent and dependent.
The independent variable is the thing that we change on purpose to see what happens. It is usually the cause or the input of the situation. For example, in the ramp experiment, the independent variable is the height of the ramp. We can choose different heights and measure how fast the car goes.
The dependent variable is the thing that we measure or observe to see the effect of the independent variable. It is usually the result or the output of the situation. For example, in the ramp experiment, the dependent variable is the speed of the car. We can use a stopwatch or a speedometer to record how fast the car goes for each height.
To tell the difference between the independent and dependent variable in context, we can ask ourselves two questions:
- Which variable are we changing on purpose?
- Which variable are we measuring or observing?
The variable that we are changing on purpose is the independent variable. The variable that we are measuring or observing is the dependent variable. Sometimes, it helps to write the variables as a sentence with the word "depends" in it. For example, the speed of the car depends on the height of the ramp. This tells us that the speed is the dependent variable and the height is the independent variable.
Remember, the independent and dependent variables can change depending on the context and the question we are trying to answer. For example, if we want to know how the weight of the car affects the speed, then the weight is the independent variable and the speed is the dependent variable. The height of the ramp is no longer relevant in this case.
Understanding the difference between the independent and dependent variable can help us design better experiments, make better graphs, and interpret data more accurately.
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