Have you ever tried to build a deck? Take a look at this situation.
Ms. Layne wants to build a rectangular deck in her back yard. She wants the length of the deck to be exactly 9 feet. She wants the perimeter of her deck to be, at most, 28 feet. The perimeter of any rectangle can be found by using the expression
Write an inequality that could be used to represent
This Concept will teach you how to use the distributive property to solve inequalities. It is exactly what you will need to figure out the deck dilemma.
The inequalities that you will see in this Concept involve parentheses. We can simplify an equation with parentheses by using the distributive property. We can do this with inequalities as well. Using the distributive property can help you to simplify an inequality so that it is easier to solve.
Apply the distributive property to the left side of the inequality. Multiply each of the two numbers inside the parentheses by -9 and then add those products.
Now, solve as you would solve any two-step inequality. Since -27 is added to
The solution is
First, we use the distributive property to multiply one-half with both of the terms inside the parentheses.
Next, we subtract two from both sides of the inequality.
Now we can multiply both sides by the reciprocal of one-half which will cancel out the one-half leaving our variable alone. This is an example of the multiplicative inverse property.
The answer is that
Now let's go back to the dilemma at the beginning of the Concept.
Consider part a first.
You know that the length is 9 feet, so substitute 9 for
Since she wants the perimeter to be “at most” 28 feet, you should use the “less than or equal to”
So, this problem can be represented by the inequality
Next, consider part b.
To find all the possible values of
Now, solve as you would solve any two-step inequality. First, subtract 18 from both sides of the inequality.
Next, divide both sides of the inequality by 2. Since you are dividing by a positive number, the inequality symbol should stay the same.
The value of
Since 6 is greater than, not less than, 5, it is not a possible value of
- a mathematical statement where one quantity can be less than, greater than or equal to another quantity.
- Inverse Operation
- the opposite operation. Inverse operations are used to solve equations.
- Like Terms
- terms in an equation or inequality that have common variables or no variables.
- Distributive Property
- a term outside of a set of parentheses can be multiplied by each of the terms inside the parentheses. This simplifies the parentheses.
Here is one for you to try on your own.
First, we should apply the distributive property to the left side of the inequality. We can multiply each of the two numbers inside the parentheses by -2 and then add those products.
Next, we can add the like terms (-16 and 18) on the left side of the inequality. Using the commutative and associative properties to reorder the terms on the left side of the equation can make it easier to see how to do this.
Finally, we solve as you would solve any two-step inequality. Since 2 is added to
Now, we can isolate the variable,
The solution is
Directions: Solve each inequality.