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Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

−24 < 4 3 n

This figure shows the inequality n is greater than negative 18. Below this inequality is a number line ranging from negative 20 to negative 16 with tick marks for each integer. The inequality n is greater than negative 18 is graphed on the number line, with an open parenthesis at n equals negative 18, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 18 comma infinity, parenthesis.

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Solve the inequality t −2 8 , graph the solution on the number line, and write the solution in interval notation.

Solution

.
Multiply both sides of the inequality by −2 .
Since −2 < 0 , the inequality reverses.
.
Simplify. .
Graph the solution on the number line. .
Write the solution in interval notation. .
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Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

k −12 15

This figure shows the inequality k is greater than or equal to negative 180. Below this inequality is a number line ranging from negative 181 to negative 177 with tick marks for each integer. The inequality k is greater than or equal to negative 180 is graphed on the number line, with an open bracket at n equals negative 180, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, negative 180 comma infinity, parenthesis.

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Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

u −4 −16

This figure shows the inequality u is less than or equal to 64. Below this inequality is a number line ranging from 62 to 66 with tick marks for each integer. The inequality u is less than or equal to 64 is graphed on the number line, with an open bracket at u equals 64, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 64, bracket.

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Solve inequalities that require simplification

Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.

Solve the inequality 4 m 9 m + 17 , graph the solution on the number line, and write the solution in interval notation.

Solution

.
Subtract 9 m from both sides to collect the variables on the left. .
Simplify. .
Divide both sides of the inequality by −5, and reverse the inequality. .
Simplify. .
Graph the solution on the number line. .
Write the solution in interval notation. .
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Solve the inequality 3 q 7 q 23 , graph the solution on the number line, and write the solution in interval notation.

This figure shows the inequality q is less than or equal to 23/4. Below this inequality is a number line ranging from 4 to 8 with tick marks for each integer. The inequality q is less than or equal to 23/4 is graphed on the number line, with an open bracket at q equals 23/4 (written in), and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 23/4, bracket.

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Solve the inequality 6 x < 10 x + 19 , graph the solution on the number line, and write the solution in interval notation.

This figure shows the inequality x is greater than negative 19/4. Below this inequality is a number line ranging from negative 7 to negative 3, with tick marks for each integer. The inequality x is greater than negative 19/4 is graphed on the number line, with an open parenthesis at x equals negative 19/4 (written in), and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 19/4 comma infinity, parenthesis.

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Solve the inequality 8 p + 3 ( p 12 ) > 7 p 28 , graph the solution on the number line, and write the solution in interval notation.

Solution

Simplify each side as much as possible. 8 p + 3 ( p 12 ) > 7 p 28
Distribute. 8 p + 3 p 36 > 7 p 28
Combine like terms. 11 p 36 > 7 p 28
Subtract 7 p from both sides to collect the variables on the left. 11 p 36 7 p > 7 p 28 7 p
Simplify. 4 p 36 > −28
Add 36 to both sides to collect the constants on the right. 4 p 36 + 36 > −28 + 36
Simplify. 4 p > 8
Divide both sides of the inequality by 4; the inequality stays the same. 4 p 4 > 8 4
Simplify. p > 2
Graph the solution on the number line. .
Write the solution in interal notation. ( 2 , )
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Solve the inequality 9 y + 2 ( y + 6 ) > 5 y 24 , graph the solution on the number line, and write the solution in interval notation.

This figure shows the inequality y is greater than negative 6. Below this inequality is a number line ranging from negative 7 to negative 3 with tick marks for each integer. The inequality y is greater than negative 6 is graphed on the number line, with an open parenthesis at y equals negative 6, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 6 comma infinity, parenthesis.

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Solve the inequality 6 u + 8 ( u 1 ) > 10 u + 32 , graph the solution on the number line, and write the solution in interval notation.

This figure shows the inequality u is greater than 10. Below this inequality is a number line ranging from 9 to 13 with tick marks for each integer. The inequality u is greater than 10 is graphed on the number line, with an open parenthesis at u equals 10, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 10 comma infinity, parenthesis.

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Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.

Solve the inequality 8 x 2 ( 5 x ) < 4 ( x + 9 ) + 6 x , graph the solution on the number line, and write the solution in interval notation.

Solution

Simplify each side as much as possible. 8 x 2 ( 5 x ) < 4 ( x + 9 ) + 6 x
Distribute. 8 x 10 + 2 x < 4 x + 36 + 6 x
Combine like terms. 10 x 10 < 10 x + 36
Subtract 10 x from both sides to collect the variables on the left. 10 x 10 10 x < 10 x + 36 10 x
Simplify. −10 < 36
The x ’s are gone, and we have a true statement. The inequality is an identity.
The solution is all real numbers.
Graph the solution on the number line. .
Write the solution in interval notation. ( −∞ , )
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Questions & Answers

integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
they are 92 candies in the bag
POP
rectangular field solutions
Navin Reply
What is this?
Donna
t
muqtaar
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from $1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
how
muqtaar
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
Kirisma Reply
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
No; 65m/hr
albert
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found $7(+7). Totally you lost just $12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
hi
albert
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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