- Math
- Middle School
- Solving Equations
- Inequality Symbols: <, >, ≤, ≥
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Eugene Lee
Basics on the topicInequality Symbols: <, >, ≤, ≥
Inequality symbols are a shorthand notation used to compare different quantities. There are four inequality symbols “greater than”, “less than”, “greater than or equal to”, and “less than or equal to”. So, for instance, the sentence “5 is greater than 2” can be written as 5>2. A good way to remember which number is greater is to think of each symbol like a mouth; the mouth will always eat the larger of the two numbers being compared. Learn about inequality symbols by helping Christopher the vampire pack the maximum number of supplies he needed for his trip to California. Common Core Reference: CCSS.MATH.CONTENT.6.EE.B.8
TranscriptInequality Symbols: <, >, ≤, ≥
Christopher the Vampire is a foodie and he needs a fresh, new story for his blog: The Vegetarian Vampire. He’s working on a new piece, so he wants to go to a place where his favorite fruit grows: the blood orange. He read on Vampedia that blood oranges grow in California, which is perfect because he’s always wanted to visit the underground gardens there. To help him pack, he uses his knowledge of inequality symbols. And he has all his supplies laid out in his bed? Capes, check. Hair gel, check. Blood orange juice, check. But how much of this stuff is he allowed to carry with him on the plane? Let's take a look at the number line.
Use of inequalities
Christopher the Vampire’s trip will last fewer than 15 days. For inequailties with 'less than', we use this sign <. Furthermore, for this trip, Chris can't take more than 1000ml of blood orange juice on the plane. For inequalities like 'less than or equal to' we use this symbol: ≤. Our foodie vampire also needs to pack more than 1 bottle of hair gel, since he ran out during his last vacation. Let's draw this on the number line. For inequalities with 'more than', we use the 'greater than' symbol. He also needs to pack at least 16 capes, one for each day and two, just in case. For inequalities with 'at least', we use the 'greater than or equal to' symbol.
Inequalities summary - Imagine a mouth
Let's take another look at the different inequality symbols. A good way to remember which number is greater, is to think of each sign like a mouth. The mouth will always eat the larger of the two numbers being compared. For example let's compare 2 and 4. Since 2 is less than 4, the mouth will eat the 4. If the mouth opens to the right, it's read: 'a' is less than 'b'. However, if the mouth opens to the left, it's read: 'a' is greater than 'b'. As we saw earlier, the greater than and less than symbols can also be combined with the equal sign. When we say 'as many as' or 'no more than', we mean 'less than or equal to' which means that a could be less than b or equal to b. But, when we say 'at least', we mean 'greater than or equal to'. Here a could be greater than b or equal to b. Let’s see how Christopher the Vampire is enjoying his vacation. OH NO! No more blood oranges?!? This might make his vacation a bit tougher...
Inequality Symbols: <, >, ≤, ≥ exercise
Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Inequality Symbols: <, >, ≤, ≥.
Explain the inequality symbols.
Hints
Here you see a number line for $>65$.
(Video) BossMaths N1e video 1 – Ordering integers, decimals and fractions using inequality symbols$\ge~$ is the same as $~>~$ including the $~=~$ relation.
Here you see a number line for $\le 55$.
Pay attention to the circle.
Solution
To distinguish the inequality symbols:
- $<~$ for the less than relation. You see the regarding number line beside. The fact, that Christopher spends less than 15 days on his trip, is represented by an empty circle.
- $\le$ for the less than or equal relation. The difference between this symbol and the $<$-symbol is the $=$ sign. This can be seen by the filled circle.
- $>$ for the greater than relation. Similar to the $<$ relation the corresponding circle is empty.
- $\ge$ for the greater than or equal relation. It also includes the $=$ sign which can be shown by a filled circle.
Identify the symbol that correctly describes the relationship.
Hints
Take a look at this example:
$4$ is more than $2$.
You can write it as $4>2$.
Or you could write it as $2<4$.
At least indicates greater than or equal.
Remember the relation signs:
- $<$ less than
- $\le$ less than or equal
- $>$ greater than
- $\ge$ greater than or equal
Solution
Christopher already knows that his trip takes fewer than 15 days. Fewer than indicates the $<$- or less than-symbol: $<15$. For the representation on a number line you use an empty circle surrounding 15.
The amount of blood orange he can take with is limited to the top by $1000~ml$, including this amount. This indicates the $\le$- or less than or equal-symbol: $\le 1000$. Here you use a filled circle.
Christopher knows how much hair gel he needs. So he concludes to pack more than one bottle of hair gel. This indicates the $>$- or greater than-symbol: $>1$. Again you use an empty circle.
Last but least he packs some capes: at least one for each day and one in spare, 16 in total. At least indicates the $\ge$- or greater than or equal-symbol: $\ge 16$. Here you use a filled circle on a number line.
But, what's that? Arriving in the blood orange garden Christopher discovers a sign: Sorry! No blood oranges.
Determine the corresponding mathematical inequality to match the number line.
Hints
This number line represents the inequality $x\le 7$.
- The arrow to the left indicates $<$ or $\le$.
- The filled circle stands for less than or equal.
This number line stands for $x>-20$.
The $\ge$-symbol can be excluded because the empty circle.
(Video) comparison of decimal numbers.दशमलव संख्याओं की तुलना। decimal numbers. Sharmila Rawat.Solution
First we consider number lines in general.
- An arrow to the left indicates a $<$ or $\le$ relation.
- An arrow to the right indicates a $>$ or $\ge$ relation.
You can decide if you have to use $<$ or $\le$ respectively $>$ or $\ge$ depending on the circle.
- An empty circle means $<$ or $>$ depending on the direction of the arrow.
- A filled circle means $\le$ or $\ge$.
So we are able to determine the corresponding inequality from left to the right:
- $x\le 8$
- $x<8$
- $x>4$
- $x\ge 4$
Examine the inequality according to different word problems.
Hints
At least indicates greater than or equal.
More than indicates the $>$-symbol.
Distinguish between less than ($<$) and less than or equal ($\le$).
Solution
You can memorize different keywords which indicate the regarding inequality-symbol:
- More than indicates greater than $>$.
- At least indicates greater than or equal $\ge$.
Speed limit Sure a speed limit doesn't indicate, that you have to drive faster than this limit. So we get the inequality $x\le 45$.
Birthday party You like to invite less than 10 friends. So we get $x<10$.
Earphones More than indicates $>$-symbol. This gives us $x>25$.
Decide which inequality symbol to use.
Hints
Pay attention
- $5<7$ but
- $-5>-7$
If you change the sign of the numbers you also have to change the inequality symbol.
(Video) Learning how to find the maximum value of an objective functionTake care of the use of $>$ or $\ge$:
- $7\ge 7$ but $7\not > 7$
- $7>4$ as well as $7\ge 4$
Solution
You can imagine the greater than symbol as a mouth.
The bigger number eats the larger one.
- For example $4>2$. You can also use the $\ge$ sign.
- Similarly $2<4$ as well as $2\le 4$. The order is changed.
Pay attention to the sign of the numbers:
- $-2>-4$ as well as $-2\ge -4$.
- The other way round we can conclude $-4<-2$ or $-4\le -2$.
Determine the corresponding inequality.
Hints
Pay attention to the circle:
- Empty circles indicate $>$ or $<$.
- Filled circles indicate $\ge$ or $\le$.
A circle indicates the number you have to use in the inequality.
The circle indicates at the one hand the number $65$ and on the other that $65$ belongs to the inequality.
The arrow to the right indicates $>$ or $\ge$.
Together we can conclude the following inequality for this number line:
$x\ge 65$.
Solution
You use number lines to represent inequalities.
First you draw a circle exactly at the position of the regarding number.
Depending on the inequality symbol the circle is filled or empty:
- Empty: $>$ or $<$
- Filled: $\ge$ or $\le$
You can decide $<$ or $\le$ respectively $>$ or $\ge$ depending on the direction of the arrow.
Here you see four different number lines, top down:
- $x>-6$
- $x\le -2$
- $x<4$
- $x\ge 2$
More videos for the topicSolving Equations
FAQs
What kind of line is used when the inequality symbol is ≥ or ≤? ›
Use a solid line if the symbol ≤ or ≥ is used because the boundary is included in the solution. Use a dashed line if < or > is used to indicate that the boundary is not part of the solution.
What does the symbol ≤ ≤ represent in an inequality? ›An inequality is a mathematical relationship between two expressions and is represented using one of the following: ≤: "less than or equal to"
When graphing inequalities The are used for the symbols ≤ and ≥? ›A closed, or shaded, circle is used to represent the inequalities greater than or equal to (≥) or less than or equal to (≤) . The end point is part of the solution. An open circle is used for greater than (>) or less than (<). The end point is not part of the solution.
When graphing an inequality on a number line if you have ≥ or ≤ the circle must be a closed circle on the number line? ›Inequalities on a number line
Open circles are used for numbers that are less than or greater than (< or >). Closed circles are used for numbers that are less than or equal to and greater than or equal to (≤ or ≥).
Greater than or Equal to (≥)
What are the 5 types of inequality symbols? ›The five inequality symbols in Maths are greater than symbol (>), less than symbol (<), greater than or equal to symbol (≥), less than or equal to symbol (≤), and not equal to symbol (≠).
How to solve an inequality? ›When solving an inequality: • you can add the same quantity to each side • you can subtract the same quantity from each side • you can multiply or divide each side by the same positive quantity If you multiply or divide each side by a negative quantity, the inequality symbol must be reversed.
What is the difference between and ≥? ›Greater than or equal to sign: ≥
a can equal b, unlike the greater than sign. This is because ≥ does not denote a strict inequality. This is the only difference between ">" and "≥".
The graph of an inequality in two variables is the set of points that represents all solutions to the inequality. A linear inequality divides the coordinate plane into two halves by a boundary line where one half represents the solutions of the inequality. The boundary line is dashed for > and < and solid for ≤ and ≥.
What is an example of linear inequality? ›What Is an Example of Linear Inequality? An example of linear inequality is x - 5 > 3x - 10. Here, the LHS is strictly greater than the RHS since greater than symbol is used in this inequality. After solving, the inequality looks like this: 2x < 5 ⇒ x < (5/2).
How do you show where an inequality is true on a number line? ›
To represent inequalities on a number line we show the range of numbers by drawing a straight line and indicating the end points with either an open circle or a closed circle. An open circle shows it does not include the value. A closed circle shows it does include the value.
What are the rules for graphing inequalities on a number line? ›To plot an inequality, such as x>3, on a number line, first draw a circle over the number (e.g., 3). Then if the sign includes equal to (≥ or ≤), fill in the circle. If the sign does not include equal to (> or <), leave the circle unfilled in.
How do you use greater than symbol in a sentence? ›This sign is used to show that one value is greater than the other value. For example, the statement "4 is greater than 2" is true. So we can write it as 4 > 2. Now, let's understand the greater than symbol from the left side and the right side.
What is the comma rule with and? ›1. Use a comma to separate independent clauses. Rule: Use a comma before a coordinating conjunction (and, but, yet, so, or nor, for) when it joins two complete ideas (independent clauses). He walked down the street, and then he turned the corner.
How do I write and or? ›Therefore, the use of the slash in “and/or” indicates that we mean “and” or also “or.” Let's look at the same example with “and/or” below. I would like a pizza and/or chips for lunch please. The slash means that I will be happy to eat a pizza, OR chips, OR both pizza AND chips for lunch.
What does 👈 👉 mean in texting? ›The majority of people agree that it means 'shy'. As if you were twiddling your fingers together, nervously. The emojis can often be paired with the emoji too, for extra nervous vibes. The emoji sequence can be used if you're about to ask someone a soft, yet risky question, or if you're just feeling hella shy.
What does *_* mean in texting? ›"In Love" is the most common definition for *_* on Snapchat, WhatsApp, Facebook, Twitter, Instagram, and TikTok. *_* Definition: In Love.
What does := mean in math? ›Pierre Bouguer (1698-1758) later refined these to ≤ (“less than or equals”) and ≥ (“greater than or equals”) in 1734. := (the equal by definition sign) means “is equal by definition to”. This is a common alternate. form of the symbol “=Def”, which appears in the 1894 book Logica Matematica by the logician.
What are the 3 different types of inequality? ›Related concepts are lifetime Inequality (inequality in incomes for an individual over his or her lifetime), Inequality of Wealth (distribution of wealth across households or individuals at a moment in time), and Inequality of Opportunity (impact on income of circumstances over which individuals have no control, such ...
How do you solve linear inequalities step by step? ›- Step 1: Solve the inequality for y. ...
- Step 2: Graph the boundary line for the inequality. ...
- Step 3: Shade the region that satisfies the inequality. ...
- Step 4: Solve the second inequality for y. ...
- Step 5: Graph the boundary line for the second inequality. ...
- Step 6: Shade the region that satisfies the second inequality.
What is an inequality formula? ›
Unit Overview. An inequality is a mathematical statement that compares algebraic expressions using greater than (>), less than (<), and other inequality symbols. A compound inequality is a pair of inequalities joined by and or or.
How do you write an inequality step by step? ›- Step 1 Eliminate fractions by multiplying all terms by the least common denominator of all fractions.
- Step 2 Simplify by combining like terms on each side of the inequality.
- Step 3 Add or subtract quantities to obtain the unknown on one side and the numbers on the other.
There are four ways to represent an inequality: Equation notation, set notation, interval notation, and solution graph.
What is the difference of ≤? ›Two other comparison symbols are ≥ (greater than or equal to) and ≤ (less than or equal to).
What is the difference between and ≤? ›The < symbol is “less than”, and = is “equal to”. Put them together, you have ≤ which is “less than or equal to”. It's a comparison operator.
What is the difference between () and [] in math? ›The notation may be a little confusing, but just remember that square brackets mean the end point is included, and round parentheses mean it's excluded. If both end points are included the interval is said to be closed, if they are both excluded it's said to be open.
Which is a linear inequality in 2 variables? ›A linear inequality in two variables is formed when symbols other than equal to, such as greater than or less than are used to relate two expressions, and two variables are involved.
How do you find the boundary line for a linear inequality? ›Let's graph the inequality x + 4y ≤ 4. To graph the boundary line, find at least two values that lie on the line x + 4y = 4. You can use the x- and y- intercepts for this equation by substituting 0 in for x first and finding the value of y; then substitute 0 in for y and find x.
How do I solve a linear equation? ›To solve linear equations, find the value of the variable that makes the equation true. Use the inverse of the number that multiplies the variable, and multiply or divide both sides by it. Simplify the result to get the variable value. Check your answer by plugging it back into the equation.
How do you write a linear equation as an inequality? ›Linear equations use the equal sign ( =). Linear inequalities use inequality signs ( > , < , ≥is greater than or equal to, and ≤is less than or equal to).
Which inequality is and or? ›
Compound Inequalities: And/Or Inequalities
A compound inequality is made up of two inequalities connected by the word 'and' or the word 'or.
If the inequality is a “>” or “<”, then the graph will be an open half‐plane. An open half‐plane does not include the boundary line, so the boundary line is written as a dashed line on the graph.
How do you know if it's a solid or dotted line? ›If points along the boundary line are included in the solution set, then a solid line is used; if points along the boundary line are not included then a dotted line is used. divides the plane into two half planes. In this case, it is shown as a dashed line, as the points on the line don't satisfy the inequality.
What are the 5 inequality symbols? ›The five inequality symbols in Maths are greater than symbol (>), less than symbol (<), greater than or equal to symbol (≥), less than or equal to symbol (≤), and not equal to symbol (≠).
Which inequality symbols mean `` greater than or equal to? ›The greater than symbol is >. So, 9>7 is read as '9 is greater than 7'. The less than symbol is <. Two other comparison symbols are ≥ (greater than or equal to) and ≤ (less than or equal to).
How do you solve a linear inequality system? ›- Step 1: Solve the inequality for y. ...
- Step 2: Graph the boundary line for the inequality. ...
- Step 3: Shade the region that satisfies the inequality. ...
- Step 4: Solve the second inequality for y. ...
- Step 5: Graph the boundary line for the second inequality. ...
- Step 6: Shade the region that satisfies the second inequality.
The method of solving linear inequalities in two variables is the same as solving linear equations. For example, if 2x + 3y > 4 is a linear inequality, then we can check the solution, by putting the values of x and y here. Since, 8 > 4, therefore, the ordered pair (1, 2) satisfy the inequality 2x + 3y > 4.
What line should be drawn if the inequality symbol is less than or less than or equal to? ›When graphing a linear inequality on a number line, use an open circle for "less than" or "greater than", and a closed circle for "less than or equal to" or "greater than or equal to".
When graphing a linear inequality How do you determine which area to shade? ›To determine whether to shade above the line or below the line, choose a point that is not on the line and test its coordinates in the original inequality. If the coordinates of the point satisfy the inequality, then the area above or below the line, containing the point, will be shaded.