Questions on Algebra: Inequalities, trichotomy answered by real tutors!

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Let the number be N and let its digits be x (tens) and y (ones). Then,

We can create two equations from what's given,


So,


Multiply the 1st eqn by 9 and add it to the second to get,


Divide the second by 9 to get y=8. Then, we have

So,

So, the original number was 38.

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let the digits be x & y x in the tens place and y in the units place
x+y=11
The number is 10x+y
reversed
10y+x
10y+x= 10x+y +45
9y-9x=45
/9
y-x=5
x+y =11
y-x=5
add the two equations
2y=16
/2
x=8
if x = 8 then y= 3
the number is 83
m.ananth@hotmail.ca

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-18 = m - 30


-18 + 30 = m - 30 + 30


12 = m


m = 12


So the solution is m = 12

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The problem you are given to solve is:
.

.
This problem contains a contradiction. By definition the absolute value of a quantity must be either zero or positive. It cannot be less than zero (meaning it cannot be negative).
.
So by definition the left side, which entirely an absolute value, must be zero or positive. Therefore it cannot be less than zero as it is shown.
.
I hope this helps you understand the concept of absolute values a little more.

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Divide everything in the inequality by 3



x = 5 works since 5^3 = 125. 4 and 6 don't satisfy so the only solution is x = 5.

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x in an integer such that 200<3x^3<400, find x.
-----
Divide thru by "3" to get:
(200/3) < x^3 < (400/3)
----
Take the cube root along the line:
---
cbrt[200/3] < x < cbrt[400/3]
4.0548 < x < 5.1087
---
Answer: 5
==================================
cheers,
Stan H.

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Write the following in inequality notation:
2 x + 5 < 5 x -1
add -5x to both sides
-3x+5<-1
add -5 to both sides
-3x<-6
divide both sides by -3 and reverse inequality sign
x>2
solution:
(2,∞)

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Do you mean

(which cannot really be solved algebraically), or

, or



or even ? Make sure you check for errors or ambiguities before you post.

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2x + 4 > 1/3
subtract 4 from both sides of the equation to get:
2x > 1/3 - 4
multiply 4 by 3/3 to get:
2x > 1/3 - 12/3
simplify to get:
2x > -11/3
divide both sides of the equation by 2 to get:
x > -11/(3*2)
simplify to get:
x > -11/6
since this can't be simplified any further, that' your answer.
confirm by substituting for x in your original equation.
pick a value of x > -11/6
pick a value for x that is greater than -11/6.
try -10/6
your original equation is:
2x + 4 > 1/3
substitute -10/6 for x to get:
2 * (-10/6) + 4 > 1/3
simplify to get:
-20/6 + 4 > 1/3
multiply 4 by 6/6 to get:
-20/6 + 24/6 > 1/3
combine like terms to get:
4/6 > 1/3
simplify to get:
2/3 > 1/3
since this equation is true, your value for x is probably good.
to make a complete test, you would need to test out when x = -11/6 and when x < -11/6.
both those tests should lead to a false equation (one that is not true).
for example:
when x = -11/6, the original equation of:
2x + 4 > 1/3 becomes:
2 * (-11/6) + 4 > 1/3
simplify to get:
-22/6 + 4 > 1/3
multiply 4 by 6/6 to get:
-22/6 + 24/6 > 1/3
combine like terms to get:
2/6 > 1/3
simplify to get:
1/3 > 1/3
this equation is false because 1/3 is equal to 1/3 and not greater than it.
the value of x = -11/6 is not valid.
this is correct because the solution to the equation said that x > -11/6, not x = to -11/6 and not x >= -11/6.

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"the sum of 15 and n is less than 8" translates to









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Let be the number of $1 bills;
let be the number of $5 bills, and
Let be the number of $10 bills, as suggested
We start with equations, because some numbers seem to be equal.
We can translate "twice as many tens as fives" as

We can translate "four times as many ones than fives" as
, although I would have stated that as "four times as many ones as fives."
The total amount of money would be $, but we can substitute the expressions above to have everything as a function of , the number of $5 bills.
$=$=$=$
So we have $.
We know that the amount is "more than $75 but less than $120," so
<--- That's two inequalities in one, but it's OK to write it like this (at least in my book).
So we divide all 3 sides of the inequalities by 29 and get an equivalent double inequality.

is approximately 2.586 and is approximately 4.138, but is suposed to be a natural number (a positive integer).
So it could only be , or .
If , the amount of money you have is $=$=$87.
If , the amount of money you have is $=$=$116.

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-2/3f+3 < -9

-2/3f+3-3 < -9-3

-2/3f < -12

-2f < -12*3

-2f < -36

f > -36/(-2)

f > 18
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Remember "PEMA"

Parentheses, Exponents, Multiplication, Addition.   [Remember that division is
a special kind of multiplication, and that subtraction is a special kind of
addition.  Some people use "PEMDAS", but that sometimes makes students think
that multiplication comes before division and that addition comes before
subtraction, when it doesn't.  That is determined by left takes precedence over
right]

Just do one operation at a time, and copy everything else over and don't skip
steps, and you'll always get the right answer.

7[-6+5(-3+5)]

First we look for Parentheses.  The innermost parentheses is (-3+5), so we
replace that by (2) and copy everything else over:

7[-6+5(2)]

We look again for parentheses, the innermost one is now [-6+5(2)], and within 
that parentheses we look for a parentheses.  We find (2) but there is nothing
to do in that parentheses, so we go next to exponents within that set of
parentheses [-6+5(2)].  There are no exponents in there.  So we look for
multiplication and we see 5(2) which we replace by 10 and copy everything else
over:

7[-6+10]

We look again for parentheses, the innermost one is now [-6+10], and within 
that parentheses we look for a parentheses.  Within that there is only -6+10 to
do, so we replace [-6+10] by [4] and copy everything else over:

7[4]

Finally we multiply 7 by 4 and get 28

Answer: 28

Edwin

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Solve the inequality: 3x + 2 ≤ 1
-------
Subtract 2, then divide by 3

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Solve the inequality - 4x < -7
----------
Divide by 4
-x < -7/4
Divide by -1, change the < to >
x > 7/4
===========
Or you can view it this way:
- 4x < -7
Add 4x to both sides
0 < 4x - 7
Add 7
7 < 4x
Divide by 4
7/4 < x
x > 7/4

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Solve the compound inequality. 3(x-4)<9 or x+1>5
**
3(x-4)<9
3x-12<9
3x<21
x<7
..
x+1>5
x>4
..
solution:
4 < x < 7
or
(4,7)

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subtract from both sides

Divide both sides by and
reverse the inequality sign



--------------
check answer:
Let ( It is less than -(29/4) )





OK

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x^2+4x+x+4=
x^2+5x+4

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Start with the given inequality.


Add to both sides.


Combine like terms on the right side.


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Answer:

So the solution is
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The inequality is where 'h' is the person's height in inches.

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|2x+5|-1 < 6

|2x+5|-1+1 < 6+1

|2x+5| < 7

-7 < 2x+5 < 7

-7-5 < 2x+5-5 < 7-5

-12 < 2x < 2

-12/2 < 2x/2 < 2/2

-6 < x < 1

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If 'N' is the number of games and N(n)=n^2-n, then N(7)=7^2-7=42.

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34 > 4 - 6x
Add 6x to both sides
6x + 34 > 4
Subtract 34 from both sides
6x > 4 - 34
6x > -30
divide both sides by 6
x >
x > -5
:
Check solution x > -5, let's say x = -4
34 > 4 - 6(-4)
34 > 4 + 24
34 > 28; which is true
:
However if you try it with x = -6, it will not be true as you would expect

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the product of the page numbers on 2 facing pages of a book is 156. find the page numbers.
------------
Factor 156, you'll see it has a 13.
You can make up some equations, but then you have to factor it anyway.
Or use the quadratic eqn.

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3/10(e-2) is less than or equal to 3/5e - 1/4
**
First, I will write the equation I believe is the given problem:
(3/10)(e-2)≤(3/5)e-1/4
3(e-2)/10≤3e/5-1/4
LCD:20 (like you said)
6(e-2)≤12e-5 (yes, you should multiply 6*(e-2))
6e-12)≤12e-5
-6e≤7
Divide by -6 and reverse inequality sign
e≥-7/6
solution in interval notation
[-7/6,∞)

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I need help with inequalities.
"Write the following inequality in slope-intercept form."
4x-2y>-24
-----
Add 2y to both sides to get:
4x > 2y -24
---
Add 24 to both sides to get:
2y < 4x + 24
----
Divide both sides by 2 to get the slope-intercept form:
y < 2x + 12
=======================
Cheers,
Stan H.
=======================

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|(1/x)-3| > 6
---
The problem has 2 parts:
(1/x)-3 > 6 ; (1/x)-3 < -6
------------------------------
(1/x) > 9 ; (1/x) < -3
Notice x must equal 1/10, 1/11, 1/12 etc.(getting closer and closer to zero)
Notice x must equal -1/4, -1/5 , -1/6 etc.(getting closer and closer to zero)
----
-1/3< x < 0 ; 0< x < 1/9
----
Solution: (-1/3 , 0) U (0 , 1/9)
=============================================
Cheers,
Stan H.

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y>-5x+10
Draw the boundary as a dashed line:
Plot (0,10) and (2,0) and draw a dashed line thru them:
Shade the half-plane ABOVE the boundary because you
want y GREATER than -5x+10.
----------------------------------------------------
y<-2x+15
Draw the boundary as a dashed line:
Plot (0.15) and (2,11) and draw the boundary.
Shade the half-plane BELOW the boundary because you
want y LESS than -2x+15
===================================
Cheers,
Stan H.

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Cross-multiply



Cancel out x^2, add 4 to both sides

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We have (x+4)(x+9) > 0. What you would do is test certain values of x to see which intervals satisfy. For example, suppose x = -5. Then (-5+4)(-5+9)>0, a false statement. Hence the answer is (infinity, -9) U (-4, infinity).

The other method to do is it note that (x+4)(x+9) is a parabola that opens upward, so the part between the roots -4, -9 would be below the x-axis.

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x+10<-2 or x+10>4


x+10-10<-2-10 or x+10-10>4-10


x < -12 or x > -6


So the solution is x < -12 or x > -6 which graphs to




Note:
There is an open circle at which means that we're excluding that value from the solution set.


Also, there is an open circle at which means that we're excluding that value from the solution set.
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Start with the given inequality.


Distribute.


Combine like terms on the left side.


Add to both sides.


Add to both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate . note: Remember, the inequality sign flips when we divide both sides by a negative number.


Reduce.


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Answer:

So the solution is
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Replace the value into the inequality. If you obtain a true statement, that value is a valid solution.

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-(10x-1)>96 | Distribute the -1, in other words change the sign of the terms inside ().

-10x + 1 > 96 | subtract 1 from both sides
-10x>95 | Divide by -10. If dividing by a - with an inequality, switch the inequality

x<-9.5

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