Transcript any real number - Lamar County School District

Bellwork
1. What is a solution to an equation or inequality?
2. Determine whether 2/3 is a solution to this
equation: 3x - 10 = -8.
3. Determine if -1/2 is a solution to 4x – 3 < -12 ?
Solving Linear Equations
and Inequalities
Section 1.3/1.6
Vocabulary



Equation: a mathematical sentence
that contains an equal sign.
Inequality: a mathematical
sentence that contains an
inequality symbol.
Solution: a number that makes the
sentence true.
To solve an
Equation/Inequality:



Isolate the variable!
Don't forget to flip the inequality symbol if you
multiply or divide by a negative
Always write the variable on the left when
finished with the problem.
Find X!
3 Types of Solutions to Equations
1. The equation has ONE REAL solution
(which will be x = #)
2. The equation has ANY REAL NUMBER
as a solution
(which will be 0 = 0)
3. The equation has NO REAL solution
(which will be 0 = #)
Solve each, then check your solution
1. 3/7 x + 9 = 15
-9 -9
3/7x = 6
7/3(3/7x) = 6 (7/3)
x = 14
2. 5n + 11 = 7n - 9
- 11
- 11
5n = 7n -20
-7n -7n
-2n = -20
-2 -2
n = 10
Solve each, then check your solution
3.
4(3x - 5) = -2(-x + 8) - 6x
12x - 20 = 2x - 16 - 6x
12x - 20 = - 4x - 16
+ 20
+ 20
12x = -4x + 4
+4x +4x
16x = 4
16
16
4. 1/3x + 1/4 = X - 1/6
12(1/3x + 1/4 = X - 1/6)
4X + 3 = 12X - 2
-3
-3
4X = 12X - 5
-12X -12X
x = 1/4
-8X = -5
-8
-8
X = 5/8
Solve each, then check your solution
5. -4(6x - 5) = 23 - 3(8x + 1)
-24x + 20 = 23 – 24x – 3
-24x + 20 = 20 – 24x
- 20
-20
-24x = -24x
+ 24x
+24x
0=0
All real numbers
6. -2x - (x + 3) = 3x – 3
-2x – x – 3 = 3x – 3
-3x – 3 = 3x – 3
+3
+3
-3x = 3x + 0
-3x -3x
-6x = 0
-6
-6
X=0
Solve each, then check your solution
7. -2/3(6/5x - 7/10) = 17/20
-12/15x + 7/15 = 17/20
60(-12/15x + 7/15 = 17/20)
-48x + 28 = 51
-28
-28
-48x = 23
-48
-48
X = - 23/48
8. 2(x - 7) + 3 = 2x – 14
2X – 14 + 3 = 2x –14
2X – 11 = 2x – 14
+ 11
+ 11
2X = 2x – 3
-2x
-2x
0 = -3
No Solution!
Without solving...
Tell whether the equation below has no solution, one
solution, or any real solution, then explain.
ARN, 0=0
a) 5x + 2 = 5x + 2 _____________
One, X = #
b) 5x + 2 = 2 – 5x _____________
No Sol, 0 = #
c) 3x – 2 = 3x _____________
No Sol, 0 = #
d) 2(x – 7) + 3 = 2x + 14 _____________
ARN, 0 = 0
e) 3(2x – 1) = 6x – 3 _____________
Types of Solutions to
Inequalities
Set Notation Interval
Notation
X<#
(-∞ , # )
X<#
(-∞ , # ]
X>#
(#,∞)
X>#
(#,∞ ]
Graph
Solve and graph each on a number line. Put
the answers in interval notation.
1. 5y – 8 < 12
5Y < 20
Y<4
(-∞, 4)
2. 2x + 1 < 6x – 1
-4x < -2
X>½
[ ½,∞)
Solve and graph each inequality. Write the solution in
interval notation.
3. 3 < 2/3 x + ½
18 < 4x + 3
15 < 4x
15/4 < x
X > 15/4
( 15/4, ∞)
4. 7x + 9 > 10x – 12
-3x > -21
X<7
(-∞, 7]
Solve and graph each inequality. Write the solution in
interval notation.
5. 1/3(5/8x + 2) > 2/3 x + 2 4. 4(1/2x + 7/3) > 1/3(x + 12)
5/24x + 2/3 > 2/3x + 2
2x + 28/3 > 1/3x + 4
24[5/24x + 2/3 > 2/3x + 2]
3[2x + 28/3 > 1/3x + 4]
5x + 16 > 16x + 48
6x + 28 > x + 12
-11X > 32
5X > -16
x < -32/11
X > -16/5
( -∞, -32/11]
(-16/5, ∞)
Word Problems!
salary of $16,000 plus 12% of the total value of the
appliances he sells for the year (commission). Last
year he earned $18,800. What was the total value of
the appliances he sold?
Coach Barr has decided to put a
walking track of uniform width
around the outside of the football
field and then fence in the entire
area. The football field is 160 ft
by 300 ft and he has 968 ft of
fencing to put around the entire
area (football field AND walking
track). What is the maximum
width of the walking track?
1. A real estate broker's salary
is $18,000. She earns a 4%
commission on total sales.
How much must she sell to
earn $55,000 total.
2. You plan to frame a 5 in
by 7 in photo. The frame
will have a uniform width.
The perimeter of the framed
photo will be 4o in. What is
the width of the frame?
3. A restaurant server
earns a base salary of $6
per hour plus tips. If he
averages $12 per hour in
tips, how many hours must
he work to ear a total of
$333?
4. The formula to convert
Fahrenheit to Celsius is F
= 9/5 C + 32. If the
temperature outside is 45
degrees Fahrenheit, what is
the temperature in Celsius?
5. A garden is 3 feet by 5 feet. If you want to
put a sidewalk around the garden that is
uniform width, what is the width of the sidewalk
if the perimeter of the garden and sidewalk is
28 feet ?