2.7 Solve linear inequalities (Page 2/8)

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The inequality $x \leq 1$ means all numbers less than or equal to 1. There is no lower end to those numbers. We write $x \leq 1$ in interval notation as $(− \infty, 1]$ . The symbol $− \infty$ is read as ‘negative infinity’. [link] shows both the number line and interval notation.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 1, bracket. — The inequality $x \leq 1$ is graphed on this number line and written in interval notation.

Inequalities, number lines, and interval notation

This figure show four number lines, all without tick marks. The inequality x is greater than a is graphed on the first number line, with an open parenthesis at x equals a, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, a comma infinity, parenthesis. The inequality x is greater than or equal to a is graphed on the second number line, with an open bracket at x equals a, and a red line extending to the right of the bracket. The inequality is also written in interval notation as bracket, a comma infinity, parenthesis. The inequality x is less than a is graphed on the third number line, with an open parenthesis at x equals a, and a red line extending to the left of the parenthesis. The inequality is also written in interval notation as parenthesis, negative infinity comma a, parenthesis. The inequality x is less than or equal to a is graphed on the last number line, with an open bracket at x equals a, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma a, bracket.

Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in [link] .

This figure shows the same four number lines as above, with the same interval notation labels. Below the interval notation for each number line, there is text indicating how the notation on the number lines is similar to the interval notation. The first number line is a graph of x is greater than a, and the interval notation is parenthesis, a comma infinity, parenthesis. The text below reads: “Both have a left parenthesis.” The second number line is a graph of x is greater than or equal to a, and the interval notation is bracket, a comma infinity, parenthesis. The text below reads: “Both have a left bracket.” The third number line is a graph of x is less than a, and the interval notation is parenthesis, negative infinity comma a, parenthesis. The text below reads: “Both have a right parenthesis.” The last number line is a graph of x is less than or equal to a, and the interval notation is parenthesis, negative infinity comma a, bracket. The text below reads: “Both have a right bracket.” — The notation for inequalities on a number line and in interval notation use similar symbols to express the endpoints of intervals.

Graph on the number line and write in interval notation.

ⓐ $x \geq −3$ ⓑ $x < 2.5$ ⓒ $x \leq - \frac{3}{5}$

Solution

ⓐ

Shade to the right of $−3$ , and put a bracket at $−3$ .

Write in interval notation.
ⓑ

Shade to the left of $2.5$ , and put a parenthesis at $2.5$ .

Write in interval notation.
ⓒ

Shade to the left of $- \frac{3}{5}$ , and put a bracket at $- \frac{3}{5}$ .

Write in interval notation.

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Graph on the number line and write in interval notation:

ⓐ $x > 2$ ⓑ $x \leq - 1.5$ ⓒ $x \geq \frac{3}{4}$

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Graph on the number line and write in interval notation:

ⓐ $x \leq - 4$ ⓑ $x \geq 0.5$ ⓒ $x < - \frac{2}{3}$

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Solve inequalities using the subtraction and addition properties of inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.

Properties of equality

\begin{matrix} Subtraction Property of Equality & Addition Property of Equality \\ For any numbers a, b, and c, & For any numbers a, b, and c, \\ \begin{matrix} if & a & = & b, \\ then & a - c & = & b - c . \end{matrix} & \begin{matrix} if & a & = & b, \\ then & a + c & = & b + c . \end{matrix} \end{matrix}

Similar properties hold true for inequalities.

For example, we know that −4 is less than 2.
If we subtract 5 from both quantities, is the left side still less than the right side?
We get −9 on the left and −3 on the right.
And we know −9 is less than −3.
	The inequality sign stayed the same.

Similarly we could show that the inequality also stays the same for addition.

This leads us to the Subtraction and Addition Properties of Inequality.

Properties of inequality

\begin{matrix} Subtraction Property of Inequality & Addition Property of Inequality \\ For any numbers a, b, and c, & For any numbers a, b, and c, \\ \begin{matrix} if & a & < & b \\ then & a - c & < & b - c . \\ if & a & > & b \\ then & a - c & > & b - c . \end{matrix} & \begin{matrix} if & a & < & b \\ then & a + c & < & b + c . \\ if & a & > & b \\ then & a + c & > & b + c . \end{matrix} \end{matrix}

We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality $x + 5 > 9$ , the steps would look like this:

\begin{matrix} x + 5 & > & 9 \\ Subtract 5 from both sides to isolate x . & x + 5 - 5 & > & 9 - 5 \\ Simplify. & x & > & 4 \end{matrix}

Any number greater than 4 is a solution to this inequality.

Solve the inequality $n - \frac{1}{2} \leq \frac{5}{8}$ , graph the solution on the number line, and write the solution in interval notation.

Solution


Add $\frac{1}{2}$ to both sides of the inequality.
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.

$p - \frac{3}{4} \geq \frac{1}{6}$

This figure shows the inequality p is greater than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at p equals 11/12, and a dark line extends to the right from 11/12. Below the number line is the solution written in interval notation: bracket, 11/12 comma infinity, parenthesis.

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

$r - \frac{1}{3} \leq \frac{7}{12}$

This figure shows the inequality r is less than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at r equals 11/12, and a dark line extends to the left from 11/12. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/12, bracket.

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Solve inequalities using the division and multiplication properties of inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

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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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Shade to the right of $−3$ , and put a bracket at $−3$ .
Write in interval notation.


Shade to the left of $2.5$ , and put a parenthesis at $2.5$ .
Write in interval notation.


Shade to the left of $- \frac{3}{5}$ , and put a bracket at $- \frac{3}{5}$ .
Write in interval notation.