2.7 Solve linear inequalities (Page 3/8)

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Properties of equality

\begin{matrix} Division Property of Equality & Multiplication Property of Equality \\ For any numbers a, b, c, and c \neq 0, & For any real numbers a, b, c, \\ \begin{matrix} if & a & = & b, \\ then & \frac{a}{c} & = & \frac{b}{c} . \end{matrix} & \begin{matrix} if & a & = & b, \\ then & a c & = & b c . \end{matrix} \end{matrix}

Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?

Consider some numerical examples.


Divide both sides by 5.		Multiply both sides by 5.
Simplify.
Fill in the inequality signs.

The inequality signs stayed the same.

Does the inequality stay the same when we divide or multiply by a negative number?


Divide both sides by −5.		Multiply both sides by −5.
Simplify.
Fill in the inequality signs.

The inequality signs reversed their direction.

When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.

Here are the Division and Multiplication Properties of Inequality for easy reference.

Division and multiplication properties of inequality

\begin{matrix} For any real numbers a, b, c \\ \begin{matrix} if a < b and c > 0, then \frac{a}{c} < \frac{b}{c} and a c < b c . \\ if a > b and c > 0, then \frac{a}{c} > \frac{b}{c} and a c > b c . \\ if a < b and c < 0, then \frac{a}{c} > \frac{b}{c} and a c > b c . \\ if a > b and c < 0, then \frac{a}{c} < \frac{b}{c} and a c < b c . \end{matrix} \end{matrix}

When we divide or multiply an inequality by a:

positive number, the inequality stays the same .
negative number, the inequality reverses .

Solve the inequality $7 y < 42$ , graph the solution on the number line, and write the solution in interval notation.

Solution


Divide both sides of the inequality by 7. Since $7 > 0$ , the inequality stays the same.
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.

$9 c > 72$

$c > 8$
This figure is a number line ranging from 6 to 10 with tick marks for each integer. The inequality c is greater than 8 is graphed on the number line, with an open parenthesis at c equals 8, and a dark line extending to the right of the parenthesis.

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

$12 d \leq 60$

$d \leq 5$
This figure is a number line ranging from 3 to 7 with tick marks for each integer. The inequality d is less than or equal to 5 is graphed on the number line, with an open bracket at d equals 5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 5, bracket.

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Solve the inequality $−10 a \geq 50$ , graph the solution on the number line, and write the solution in interval notation.

Solution


Divide both sides of the inequality by −10. Since $−10 < 0$ , the inequality reverses.
Simplify.
Graph the solution on the number line.
Write the solution in interval notation.

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

$−8 q < 32$

$q > −4$
This figure is a number line ranging from negative 6 to negative 3 with tick marks for each integer. The inequality q is greater than negative 4 is graphed on the number line, with an open parenthesis at q equals negative 4, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 4 comma infinity, parenthesis.

Got questions? Get instant answers now!

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

$−7 r \leq - 70$

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

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Solving inequalities

Sometimes when solving an inequality, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable to the left.

\begin{matrix} x > a has the same meaning as a < x \end{matrix}

Think about it as “If Xavier is taller than Alex, then Alex is shorter than Xavier.”

Solve the inequality $−20 < \frac{4}{5} u$ , graph the solution on the number line, and write the solution in interval notation.

Solution


Multiply both sides of the inequality by $\frac{5}{4}$ . Since $\frac{5}{4} > 0$ , the inequality stays the same.
Simplify.
Rewrite the variable on the left.
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.

$24 \leq \frac{3}{8} m$

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

Aislinn Reply

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what is physics

Siyaka Reply

A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance

Jude Reply

Can you compute that for me. Ty

Jude

what is the dimension formula of energy?

David Reply

what is viscosity?

David

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what is chemistry

Youesf Reply

what is inorganic

emma

Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes

Adjei

please, I'm a physics student and I need help in physics

Adjanou

chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.

Pedro

A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity

Krampah Reply

2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

Sahid Reply

you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

Samuel Reply

can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

Joseph Reply

Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

Joseph

Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

Joseph

"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

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Maurice Reply

what are the types of wave

Maurice

answer

Magreth

progressive wave

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Mujahid

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

yasuo Reply

Who can show me the full solution in this problem?

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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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