Informative line

Inequality As Constraint

Inequality

  • An inequality says that the two values are not equal.

\(a\neq b\) says that \(a\) is not equal to \(b.\)

  • Sometimes, we come across such situations where two quantities are not equal \((\neq)\). One may be greater than or less than the other one.

For example: \(3\neq5\)

But \(3<5\)

We use the following symbols when a math sentence is not equal:

(i) \(>\, \rightarrow\)  greater than

(ii) \(<\, \rightarrow\) less than

(iii) \(\ge\, \rightarrow\) greater than or equal to ("or equal to" part is indicated by the line underneath of the \(>\) symbol)

(iv) \(\leq \, \rightarrow\) less than or equal to ("or equal to" part is indicated by the line underneath of the \(<\) symbol)

Illustration Questions

Which one of the following shows an inequality?

A \(x+1=2\)

B \(2x+3\)

C \(x+y=4\)

D \(x>7\)

×

An inequality says that the two values are not equal, which means one may be greater than (>) or less than (<) the other one.

Option (A) is  \(x+1=2\) which represents an equation, because it contains an equals (=) sign.

Hence, option (A) is incorrect.

Option (B) is  \(2x+3\) which represents an expression, because it does not contain an equals sign.

Hence, option (B) is incorrect.

Option (C) is \(x+y=4\) which represents an equation, because it contains an equals sign.

Hence, option (C) is incorrect.

Option (D) is \(x>7\) which represents an inequality as it says that \(x\) is greater than \(7.\)

Hence, option (D) is correct.

Which one of the following shows an inequality?

A

\(x+1=2\)

.

B

\(2x+3\)

C

\(x+y=4\)

D

\(x>7\)

Option D is Correct

Writing Inequalities for Given Phrases

  • We write inequalities when we don't have any exact solution of the given problems.
  • Just like equations, we can write inequalities on the basis of the given statements.

For example:

The sum of \(x\) and \(10\) is less than \(20\).

The statement says, sum of \(x\) and \(10\).

\(\Rightarrow\;x+10\)

\(x+10\) is less than \(20\).

\(\Rightarrow\;x+10<20\)

Illustration Questions

One-third of a number increased by \(3,\) is less than \(15\). Which one of the following inequalities correctly represents the given statement?

A \(\left(\dfrac{1}{3}\right)n>3+15n\)

B \(3n-3>15\)

C \(\left(\dfrac{1}{3}\right)n+3<15\)

D \(3n+3<15\)

×

Go step by step to write the inequality on the basis of the given statement. 

The statement says that one-third of a number is increased by \(3.\)

Let the number be \(n.\)

One-third of the number is  \(\left(\dfrac{1}{3}\right)n\) which is increased by \(3\) means the addition of \(3.\)

=\(\left(\dfrac{1}{3}\right)n+3\)

According to the statement, \(\left(\dfrac{1}{3}\right)n+3\) is less than \(15\).

That means, \(\left(\dfrac{1}{3}\right)n+3<15\) 

Hence, option (C) is correct.

One-third of a number increased by \(3,\) is less than \(15\). Which one of the following inequalities correctly represents the given statement?

A

\(\left(\dfrac{1}{3}\right)n>3+15n\)

.

B

\(3n-3>15\)

C

\(\left(\dfrac{1}{3}\right)n+3<15\)

D

\(3n+3<15\)

Option C is Correct

Solving Inequalities

  • Solving an inequality means to find all the possible values that represent the solution of the given problem.
  • To solve an inequality, do the same thing on both sides of the inequality.
  • Use inverse operations to solve an inequality.

For example:

\(x+5\leq13\)

  • To solve this inequality, subtract \(5\) from both the sides.

\(\Rightarrow\;x+5-5\leq13-5\)

\(\Rightarrow\;x\leq8\)

This is the required solution.

Illustration Questions

\(2x+3>5\) Find the value of \(x\) which makes the inequality true.

A \(x>2\)

B \(x<1\)

C \(x>1\)

D \(x\leq2\)

×

Given inequality: \(2x+3>5\)

To find \(x,\,3\) and \(2\) should be removed.

Solve for \(x\);

\(2x+3>5\)

\(\Rightarrow\;2x+3-3>5-3\)

\(\Rightarrow\;2x>2\)

\(\Rightarrow\;\dfrac{2x}{2}>\dfrac{2}{2}\)

\(\Rightarrow\;x>1\)

Hence, option (C) is correct.

\(2x+3>5\) Find the value of \(x\) which makes the inequality true.

A

\(x>2\)

.

B

\(x<1\)

C

\(x>1\)

D

\(x\leq2\)

Option C is Correct

Representation of Inequality through Number Line

  • We can represent inequality on the number line to understand it clearly.
  • First, draw a number line and then place the value from where we want to represent.
  • For greater than \((>)\) and greater than or equal to \((\geq)\) inequalities, go to the right side from the value on the number line.
  • For less than \((<)\) and less than or equal to \((\leq)\) inequalities, go to the left side from the value on the number line.

Representations of inequality on the number line:

Case (i) \(x>c\)

\(x>c\) means \(x\) can take all the values greater than \(c\) but not \(c\).

The given number line shows all the possible values which are greater than \(c.\)

Case (ii) \(x<c\)

\(x<c\) means \(x\) can take all the values less than  \(c\) but not \(c\).

The number line shows all the possible values which are less than \(c.\)

In cases (i) and (ii), the blank circles on \(c\) represent that \(c\) is not included in the solution.

Case (iii) \(x\geq c\)

\(x\geq c\) means \(x\) can take all the values greater than \(c\). Also, \(c\) is included.

The number line shows all the possible values which are greater than or equal to \(c.\)

Case (iv) \(x\leq c\)

\(x\leq c\) means \(x \) can take all the values less than \(c\). Also, \(c\) is included.

The number line shows all the possible values which are less than or equal to \(c.\)

In cases (iii) and (iv), the shaded circles show that \(c\) is included in the solution.

For example:

Represent \(x\leq7\) on the number line.

\(x\leq7\) means \(x \) can take all the values less than \(7\). Also, \(7\) is included.

The line above the number line shows all the possible values for \(x.\)

The shaded circle over \(7\) shows that \(7\) is also included in the solution.

Illustration Questions

Which option correctly represents the given inequality: \(x\geq5\)?

A

B

C

D

×

Given inequality: \(x\geq5\) is in the form of \(x\geq c\).

\(x\geq5\) means \(x\)  can take all the possible values which are greater than or equal to \(5.\) 

For \(x \) greater than \(5\), we go to the right side from \(5\) on the number line.

As \(5\) is also included in the solution, so we can show it by placing a shaded circle over \(5\).

 

Now, we can represent it through a number line as shown.

image

Hence, option (B) is correct.

Which option correctly represents the given inequality: \(x\geq5\)?

A image
B image
C image
D image

Option B is Correct

Inequality of some Standard Results

  • We write inequalities when we don't have any exact solution of the given problems.
  • Just like equations, we can write inequalities according to the statements.
  • To write inequalities for word problems, first identify the keywords:

(i) At least - means greater than or equal to \((\geq)\)

(ii) Maximum or not more than - means less than or equal to \((\le)\) 

(iii) More than - means greater than \((>)\)

(iv) Less than - means less than \((<)\) 

  • To write an inequality for word problem, follow the given steps:
  1.  Try to understand what the situation says.
  2.  Find the operations involved.
  3. Identify the keyword used in the problem to determine the inequality symbol.

Illustration Questions

Let \(x\) represents any number in the set of even whole numbers greater than \(4\). Which inequality is true for all values of \(x\)? 

A \(x>0\)

B \(x<0\)

C \(x>5\)

D \(x<2\)

×

Write the inequality according to the given statement and go step by step.

The statement says that \(x\) represents any number in the set of even whole numbers greater than \(4\).

The set of even whole numbers greater than \(4\) is,

\(\{6,\;8,\;10,\;12,...\}\)

i.e. \(x\) represents \(\{6,\;8,\;10,\;12,...\}\)

Option (A) is incorrect because here \(x\) also represents \(2\) and \(4\) which are not included in the above set.

Option (B) is incorrect because whole numbers are not negative.

Option (C) is correct because \(x\) represents \(\{6,\;8,\;10,\;12,...\}\) which are even numbers greater than \(4\).

Option (D) is incorrect because whole numbers are not negative. 

Let \(x\) represents any number in the set of even whole numbers greater than \(4\). Which inequality is true for all values of \(x\)? 

A

\(x>0\)

.

B

\(x<0\)

C

\(x>5\)

D

\(x<2\)

Option C is Correct

Discrete and Continuous Solution of Inequality

  • An inequality can have infinite solutions. 
  • These solutions can be discrete or continuous. 
  • But, what are discrete and continuous solutions?

Here, we will understand them with the help of an example.

In a race, \(10\) kids participated. The length of the race track was about \(2\) miles.

After the completion of the race, it was observed that each kid covered at least \(1.3\) miles.

Here, the number of kids who covered the distance between \(1.3\) miles to \(2\) miles, represents the discrete solution, shown with red points.

Let \(x\) denote the number of kids.

Then \(x>0\) and \(x\leq 10\)

The distance covered by kids represents the continuous solution, as the distance can be in decimals, shown by a red line.

Let \(h\) be the distance covered by kids. 

Then, \(h\geq1.3\) and \(h\leq 2\)

Illustration Questions

In a school, the height of students of grade \(1\) is between \(35\) inches to \(45\) inches. Which one of the following options represents the height of students of grade \(1\)?

A

B

C

D

×

The height of students of grade \(1\) is between \(35\) inches to \(45\) inches.

The height of students is represented by red color on the number line shown. The numbers \(35\) and \(45\) are included. It shows continuous solution because height can be anywhere between \(35\) to \(45\) inches and can be in decimals.

Negative numbers are not shown on the number line because height can not be negative.

Hence, option (A) is correct. 

image

Option (B) is incorrect because it does not represent all possible heights.

Option (C) is incorrect because it represents only three heights, \(35\) inches, \(40\) inches and \(45\) inches.

Option (D) is incorrect because it represents only the heights between \(40\) and \(45\) inches.

In a school, the height of students of grade \(1\) is between \(35\) inches to \(45\) inches. Which one of the following options represents the height of students of grade \(1\)?

A image
B image
C image
D image

Option A is Correct

Illustration Questions

Olivia's class must raise at least \($ 500\) to go on a trip. The class has collected \($200\). How much amount of money, \(m\), the class still needs to raise?

A \(m<300\)

B \(m>300\)

C \(m\ge 300\)

D \(m\le300\)

×

To represent Olivia's situation, go step by step according to the statement.

Olivia's class needs to raise at least \($500\) and till now the collection is \($200\). Thus, \(m\) is the amount, still needs to be raised.

So, \(m+200\ge500\)

Subtracting \(200\) from both the sides,

\(m+200-200\ge 500-200\)

\(m\ge300\)

Thus, the amount of money still needs to be raised, should be greater than or equal to \(300\).

image

Hence, option (C) is correct.

Olivia's class must raise at least \($ 500\) to go on a trip. The class has collected \($200\). How much amount of money, \(m\), the class still needs to raise?

A

\(m<300\)

.

B

\(m>300\)

C

\(m\ge 300\)

D

\(m\le300\)

Option C is Correct

Writing Inequalities for Given Word Problems

  • We write inequalities when we don't have any exact solution of the given problem.
  • To write inequalities for word problems, identify the keywords:

(i) at least - means greater than or equal to \((\geq)\)

(ii) maximum or not more than - means less than or equal to \((\leq)\)

(iii) more than - means greater than \((>)\)

(iv) less than - means less than \((<)\)

  • To write an inequality for word problems, follow the given steps:
  1. Try to understand what the situation says.
  2.  Find the operations involved.
  3.  Identify the keyword used in the given problem to determine the inequality symbol.

For example:

A cab charges \($2\) flat rate in addition to \($0.50\) per mile. Tina has no more than \($10\) to spend on a ride.

  • To represent Tina's situation, go step by step according to the statement.
  • Let the number of miles she can travel \(=x\)

\(\therefore\) Ride charges \(=0.50x+2\)

  • She has no more than \($10\) to spend.

Here, the keyword is "no more than" i.e. \((\leq)\) 

\(\Rightarrow\;0.50x+2\leq10\)

Illustration Questions

Kerin can run at a maximum speed of \(4\) miles per hour. Which inequality represents all the speeds, \(x,\) at which Kerin can run? Given: \(x>0\)

A \(x>4\)

B \(x<4\)

C \(x\geq4\)

D \(x\leq4\)

×

To represent Kerin's situation, go step by step according to the statement.

Here, \(x\) represents the possible speeds at which Kerin can run.

The statement says that Kerin can run at a maximum speed of \(4\) miles per hour, i.e. not more than \(4\) miles per hour. It means Kerin can run less than or equal to \(4\) miles per hour.

So, the required inequality is: \(x\leq4\) 

Hence, option (D) is correct.

Kerin can run at a maximum speed of \(4\) miles per hour. Which inequality represents all the speeds, \(x,\) at which Kerin can run? Given: \(x>0\)

A

\(x>4\)

.

B

\(x<4\)

C

\(x\geq4\)

D

\(x\leq4\)

Option D is Correct

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