Unit 4, Lesson 2
Solutions to Inequalities in One Variable
Algebra 1
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A teacher is choosing between two options for a class field trip to an orchard.
Which orchard would be cheaper to visit if the class has:
To help her compare the cost of her two options, the teacher first writes the equation , and then she writes the inequality .
Graph the solution to the inequality on the number line. Be prepared to show or explain your reasoning.
To help pay for his tuition, a college student plans to work in the evenings and on weekends. He has been offered two part-time jobs: working in the guest-services department at a hotel and waiting tables at a popular restaurant.
The equation represents a possible constraint about the situation.
Here is a graph on a number line.
Put a scale on the number line so that the point marked with a circle represents the solution to the equation.
Does one job pay better if:
Be prepared to explain or show how you know.
Here are two inequalities and two graphs that represent the solutions to the inequalities.
A
B
Graph the solution to the inequality on the number line.
The equation is an equation in one variable. Its solution is any value of that makes the equation true. Only meets that requirement, so 20 is the only solution.
The inequality is an inequality in one variable. Any value of that makes the inequality true is a solution. For instance, 30 and 48 are both solutions because substituting these values for produces true inequalities. is true, as is . Because the inequality has a range of values that make it true, we sometimes refer to all the solutions as the solution set.
One way to find the solutions to an inequality is by reasoning. For example, to find the solution to , we can reason that if 2 times a value is less than 8, then that value must be less than 4. So a solution to is any value of that is less than 4.
Another way to find the solutions to is to solve the related equation . In this case, dividing each side of the equation by 2 gives . This point, where is 4, is the boundary of the solution to the inequality.
To find out the range of values that make the inequality true, we can try values less than and greater than 4 in our inequality and see which ones make a true statement.
Let's try some values less than 4:
Let's try values greater than 4:
In general, the inequality is false when is greater than or equal to 4 and true when is less than 4.
We can represent the solution set to an inequality by writing an inequality, , or by graphing on a number line. The ray pointing to the left represents all values less than 4.