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Drill #10 Solve the following absolute value equalities. Remember to solve for both cases (positive...
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Transcript of Drill #10 Solve the following absolute value equalities. Remember to solve for both cases (positive...
Drill #10
Solve the following absolute value equalities. Remember to solve for both cases (positive and negative) and check your answers.
1. |2x – 3| = 12
2. |5 + x| + 2 = 2
3. 3 |2x – 1| + 6 = – 3
Drill #11
Solve the following absolute value equalities. Remember to solve for both cases (positive and negative) and check your answers.
1. |x + 4| = 20
2. |2x – 6| + 2 = 2
3. 3 |2x – 1| + 1 = 10
1-6 Solving Inequalities
Objective: To solve inequalities and graph the solution sets.
Calorie Intake
1. Multiply your body weight w by 4.3
2. Multiply your height h by 4.7
3. Add the numbers
4. Add 655 to your result from step 3
5. Multiply your age a times 4.7
6. Subtract the product in step 5 from the expression in step 4
Maintaining body weightThe expression for optimal Calorie intake is:
4.3w + 4.7h + 655 – 4.7a
Multiply by 1.3 to find optimal intake with moderate activity
1.3 (4.3w + 4.7h + 655 – 4.7a)
How many calories do you need to consume to maintain your weight?
Trichotomy Property
Definition: For any two real numbers, a and b, exactly one of the following statements is true:
a < b a = b a > b
A number must be either less than, equal to, or greater than another number.
(1.) Addition and Subtraction Properties For Inequalities*
1. If a > b, then a + c > b + c and a – c > b – c
2. If a < b, then a + c < b + c and a – c < b – c
Note: The inequality sign does not change when you add or subtract a number from a side
Example: x + 5 > 7
(2.) Multiplication and Division Properties for Inequalities*
For positive numbers:
1. If c > 0 and a < b then ac < bc and a/c < b/c
2. If c > 0 and a > b then
ac > bc and a/c > b/c
For negative numbers:
3. If c < 0 and a < b then
ac > bc and a/c > b/c
4. If c < 0 and a > b then
ac < bc and a/c < b/c
(3.) Non-Symmetry of Inequalities*
If x > y then y < x
• In equalities we can swap the sides of our equations:
x = 10, 10 = x
• With inequalities when we swap sides we have to swap signs as well:
x > 10, 10 < x
(4.) Solving Inequalities*• solve inequalities the same way as equations
(using S. G. I. R.)EXCEPTIONS:
Change the inequality sign when you:– multiply or divide by a negative number. – swap sides (non-symmetry property)
Example #1*: -4x + 6 > 10Example #2*: 3x > 4x + 2 – x Write your solution in a solution set.
Set Builder Notation**
Definition: The solution x < -1 written in set-builder notation:
{x| x < -1}
We say, the set of all x, such that x is less than -1.
Empty Set**
Definition: The set having no members, symbolized by { } or O
When an equation has no solution, the answer is said to be null or the empty set.
Classwork
1-6 Study Guide
#1-4
Graphing inequalities*• Graph one variable inequalities on a number
line.• < and > get open circles • < and > get closed circles• For > and > the graph goes to the right. (if the variable is on the left-hand side)• For < and < the graph goes to the left. (if the variable is on the left-hand side)
Example #1*: Graph the solution to the last example
Classwork
1-5 Practice
#1-2
Writing Inequalities (#11)*
Define a variable and write an inequality for each problem then solve and graph the solution:
4. The product of 11 and a number is less than 53.
5. The opposite of five times a number is less than 321.
Test Scores
Ron’s score on the 1st three of four 100-point chemistry tests were 90, 96, and 86. What must he score on his fourth test to have an average of at least 92 for all the tests?