Unit 6: Solving Equations & inequalities
Distinguish between expressions used in the last unit Example:
and equations used in this unit. Example:
In the last unit we simplified expressions by combining like terms.
In this unit we will be solving equations by determining what value of (or any other variable) makes the equation true.
Examples: Solve using algebra tiles. Remember that the goal is to isolate the variable.
(a)
(c)
or
(d)
Unit 6: Linear Equations & Inequalities
Note: We flipped ALL tiles. This is the same as dividing both sides by -1.
(e)
or
(f)
(h)
6.1. Solving Equations by Using Inverse Operations
When the coefficients are integers we could sometimes use inspection and usually use algebra tiles but since the coefficients can be any real number we need other strategies.
Notice that an equation is a balance, and we can do whatever we want, as long as we do it to both sides – add, subtract, multiply or divide.
We isolate the variable (get it by itself) by removing one thing at a time, usually starting with what is “furthest” from the variable. We can do this by
inversing operations, to “undo” or reverse what is in the equation. Note: Addition and Subtraction are inverse operations
Multiplication and Division are inverse operations
Examples: Solve using algebra and verify by substitution
a) b)
c) d)
g) h)
i) j)
Discuss the process needed to solve the equations in # 5-7 pp. 271-272 orally
Solving Problems
Examples:
1. For each statement below, create and solve an appropriate equation to determine each number.
a) Three times a number is -3.6
b) A number divided by 4 is 1.5
2. A rectangle has length 3.7 cm and perimeter 13.2 cm.
a) Write an appropriate equation that can be used to determine the width of the rectangle.
b) Solve the equation using algebra.
c) Verify the solution.
3. #16 p. 273
4. #19 p. 274
Set pp. 272-274 #8-15, 17, 18, 20, 21, 22
6.2. Solving Equations by Using Balance Strategies
Sometimes we will have the variable on both sides. In this case we must get the variable on only one side by maintaining a balance.
Examples: Solve using algebra and verify by substitution
a) b)
c) d)
g) h)
i) j)
k) l)
Solving Problems
Examples:
1. MPI is looking for a place to hold the school leaving ceremony due to renovations that are scheduled to take place at the school. Hall A has a set fee of $100 and a charge of $15 per person attending. Hall B has a set fee of $120 and a charge of $10 per person attending. Set up and solve an appropriate equation to determine for how many people the halls would cost the same?
6.3. Introduction to Linear Inequalities
Recall the following symbols:
< is the “less than” symbol > is the “greater than” symbol
Also, ≤ is the “less than or equal” to symbol It can also mean “no more than”
≥ is the “greater than or equal” to symbol It can also mean “no less than”
Recall that an equation has one number for the solution, but an
inequality has a set of values for the solution.
For example, if a container can hold no more than 45 kg, it can be represented by the inequality and obviously we can put different masses in the container as long as it is less than or equal to 45.
Discuss the situations in investigation p. 288
Many other real life situations can be modeled in this way.
See example 1, p. 289
Note that there are two ways to write every inequality. For instance, means the same as
means the same as means the same as means the same as
Graphing Solutions
The solution to an inequality is any value of the variable that makes the inequality true. There are often too many numbers to list, so we may use a number line.
Referred to as a closed circle
Means a number is included in a solution.
If the solution consists of integers only (discrete data) we represent the integers in the solution on a graph using closed circles. We darken an arrow on one or both ends of the number line if the pattern continues.
If the solution consists of a set of real numbers
(continuous data) we use the closed circle to indicate the first or last value when that value is included in the
solution set and the rest with a darkened segment, ray or line.
Referred to as an open circle
Means a number is not included in a solution.
If the solution consists of a set of real numbers we use the open circle to indicate the first, last, or both values when that value is not included in the solution set and the rest with a darkened segment, ray or line
The open circle is not used with discrete data.
See example 3 p. 291 ●
Note that if we have a real life application we may be restricted to discrete data (integers only), and many times even to discrete positive values.
Example: Emily’s mom said she had to invite no more than 10 people to the pool party. This could be represented by g ≤ 10 and would result in the following graph.
We can also use a number line to verify solutions.
See example 2 p. 290
Complete p. 292 #3-6 orally together
Set pp. 292-293 #7-13
6.4. Solving Linear Inequalities by Using Addition and Subtraction
Begin with the inequality that we know is true, for example, .
Add 2 and then -2 to both sides of the inequality to see if the inequality sign is affected:
Notice that the inequality sign is unaffected.
To subtract 2 is the same as adding -2 to both sides. To subtract -2 is the same as adding 2 to both sides.
Conclusion: When the same number is added to or subtracted from each side of an inequality, the resulting inequality is still true.
This means that to solve an inequality we can use the same strategy we used for solving equations:
Isolate the variable by adding or subtracting the same number from both sides of the inequality.
See the comparison on the bottom p. 295
See examples 1 & 2 pp. 296-297
6.5. Solving Linear Inequalities by Using Multiplication and Division
Begin with the inequality that we know is true, for example, .
Multiply 2 and then -2 to both sides of the inequality to see if the inequality sign is affected:
Notice that when we multiplied by a negative number, to make the inequality true we must reverse the inequality sign.
Now, beginning with the same inequality, divide both sides of the inequality by 2 and then -2 to see if the inequality sign is affected:
Notice that when we divided by a negative number, to make the inequality true we must reverse the inequality sign.
Conclusion: When each side of an inequality is multiplied or divided by the same positive number, the resulting inequality is still true.
When each side of an inequality is multiplied or divided by the same negative number, the resulting inequality must be reversed to the inequality to remain true.
This means that to solve an inequality we can use the same strategy we used for solving equations, however we must be careful when multiplying or dividing by a negative number!
See examples 1-3 pp. 302-304 and Discuss p. 304 #2
Complete p. 305 #3-6 orally together
Set pp. 305-306 #7-13, 16
