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Testing Solutions for Linear Inequalities in Two Variables

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Solving Linear Inequalities

Have you ever been to the rainforest? Take a look at this dilemma.

You are going on a trip in a jeep out to the rainforest. You need to have enough fuel to get there and back. It is 115 miles one way. You can buy two different kinds of fuel for your truck, gasoline or ethanol. Gasoline gets better gas mileage at 21 mpg but ethanol is cleaner burning and gives you 17 mpg. Use a linear inequality to figure at how many gallons of each you would need.

To write an inequality, you will first need to know about them. This Concept will teach you everything that you need to know. By the end of it, you should be able to solve Kenya’s problem.

Guidance

Our task in equations has been to find any and all of the solutions—the solution set—that make the equation true. As we consider inequalities, the task remains the same. We have to find all the values that make the statement true.

Remember the inequality symbols that you have used before.

less than  < greater than  > not equal to  \ne

less than or equal to  \le greater than or equal to  \ge

You may have seen a simple inequality like “a number is less than five” or  x < 5 and pondered the infinite values that would make  x true. You have also solved inequalities like  3(x-4) \ge 7x + 10 using properties of equations.

Just as equations can be in one or two variables, so can inequalities. However, because there are five inequality signs, we must be aware of their meanings. Additionally, there are oftentimes many ways to say the same thing—“is less than” could be said “is not as much as”—although they use the same inequality symbol.

Translate the following expressions to inequalities:

  1. The sum of two numbers is more than 10.  x+y>10
  2. The difference between two numbers is at least 32.  x-y \ge 32
  3. Four less than a number is less than one-third another number.  x-4< \frac{1}{3}y
  4. Negative 5 times the sum of two numbers is not 18.  -5(x+y) \ne 18

You can see that anytime that we are talking about two numbers or two unknown numbers that we can use two variables in an equation. This allows us to express an inequality in two variables.

Just as with equations, a solution to an inequality will be the value(s) that make the inequality true. When you saw the inequality,  x<5 , you knew that 4 is a solution, 2, 0, -3, -7.3, etc.

There are infinitely many values that make the inequality true.

When we had equations in two variables like  x+y=7 , we also saw that there were infinitely many solutions. The solutions were shown as pairs of numbers, ordered pairs, because there was more than one variable, like (3, 4) or (-1, 8).

For the same reason, we will show solutions to inequalities in two variables as ordered pairs. We can find solutions by guessing and checking or by using mathematical reasoning.

Take a look at this situation.

Which solution makes this inequality a true statement?

3x+2>y

& (3, 5) && 3 \cdot 3 +2>5?  && \text{yes}\\& (-6, 0) && 3 \cdot -6+2>0? && \text{no}\\& (-1, -1) && 3 \cdot -1+2>-1? && \text{no}\\& (10, 31) && 3 \cdot 10+2>31? && \text{yes}

Here, there are two possible solutions for this inequality. Remember that oftentimes, you will have more than one solution for an inequality.

We can also solve an inequality in the same way that we would solve a linear equation.

Take a look.

4x<16

We can solve this inequality by dividing both sides by four. Then we will isolate the variable and figure out the range of values that will be solution for this inequality.

 \frac{4x}{4} &< \frac{16}{4} \\x &< 4

Any value less than four will be a solution for this inequality. We can write it as {......4}.

Multi-step inequalities can also be solved like linear equations too.

3x-2 &> 16 \\3x &> 16+2 \\3x &> 18 \\x  &> 6

Any value greater than 6 will work as a solution for this inequality. We can write it as {6.......}.

Write an inequality for each situation.

Example A

The difference between a number and seven is greater than 12.

Solution: x-7>12

Example B

Two times a number and six is less than twenty.

Solution: 2x+6<20

Example C

Three less than four times a number is greater than 40.

Solution: 4x-3>40

Now let's go back to the dilemma from the beginning of the Concept.

To write the solution, we first need to write an inequality. We use e for ethanol and g for gasoling and we write the number of miles per gallon that each gets. We want to be able to travel there and back. Since it is 115 miles per way, that is a sum of 230 miles. Here is the inequality.

21g + 17e \ge 230

Using the ordered pair (e, g) , you could have (13, 2), (12, 3), (6, 8). The first value in each pair represents the ethanol usuage, the second value represents the gasoline usuage.

Vocabulary

Inequality
A situation where quantities are not equal

Guided Practice

Here is one for you to try on your own.

Solve this inequality.

-4x-5>15

Solution

First, we want to isolate the variable.

-4x-5+5 & > 15+5 \\-4x & > 20 \\x   & > -5

The solution set is {-5......}.

Video Review

Solving Inequalities

Practice

Directions: Translate each statement into an inequality.

  1. The difference between two numbers is greater than 8.
  2. Half of one number is at least 3 times another number.
  3. A quarter the sum of two numbers is less than 15.
  4. Seven times one number plus 3 less than another is not more than -16.
  5. Six times a number is greater than negative thirty.
  6. Five times a number and six is less than or equal to 39.
  7. Twelve divided by a number is less than seven.
  8. Six times a number and two less than another number is less than or equal to -12.

Directions: Which ordered pairs make the inequalities true?

2x-5y \ge 20

  1. (10, 5)
  2. (10, 4)
  3. (-10, -10)
  4. (0, 0)

y>-4x-2

  1. (1, 1)
  2. (0, -6)
  3. (2, -9)
  4. (-1, 0)

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