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Wize High School Algebra I Textbook (Common Core) > Solving One Variable Inequalities

Representing Inequalities

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Inequalities

By now, we are familiar with equations or equalities.

Example
x+2=5x+2=5 means that the quantity x+2x+2 is equal to (exactly the same as) 55.

There are other types of relationships aside from equalities that are called inequalities:
SymbolMeaningExamplesa<ba is less (smaller) than b2<5−3<1a>ba is greater (bigger) than b5>2−3>−5a≤ba is less than or equal to b2≤3−3≤−3a≥ba is greater than or equal to b5≥2−3≥−3a≠ba is not equal tob5≠2−3≠5\begin{array}{|c|c|c|} \hline \text{Symbol}&\text{Meaning}&\text{Examples}\\ \hline a\bct{<}b& a \text{ is less (smaller) than } b& \begin{array}{c} 2<5\\ -3<1 \end{array}\\ \hline a\bct{>}b& a \text{ is greater (bigger) than } b& \begin{array}{c} 5>2\\ -3>-5 \end{array}\\ \hline a\bct{\le} b& a \text{ is less than or equal to } b& \begin{array}{c} 2\le3\\ -3\le-3 \end{array}\\ \hline a\bct{\ge} b& a \text{ is greater than or equal to } b& \begin{array}{c} 5\ge2\\ -3\ge-3 \end{array}\\ \hline a\bct{\neq} b& a \text{ is not equal to} b& \begin{array}{c} 5\neq2\\ -3\neq5 \end{array}\\ \hline \end{array}


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Reading an Inequalities

Wize Tip
If David is 170cm tall and Nat is 150cm tall, we can say "David is taller than Nat" or "Nat is shorter than David".

When you see the inequalities 1<21<2, we can read it as
  • "1 is less than 2" OR
  • "2 is greater than 1"

How can you read the inequalities x≥3x\ge3?
  • "x is greater than or equal to 3" OR
  • "3 is less than or equal to x"

Practice: Inequalities

Select ALL possible symbols that can be placed in the following expressions.
44 ___ 1010
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Representing (Expressing) Inequalities

There are three ways to express an inequalities: verbally, algebraically, and graphically.

Expressing an Inequality Verbally

We can describe the solutions to an inequality with words such as "less than", "greater than", "less than or equal to", "greater than or equal to" or "not equal to".

Examples
  • xx is less than or equal to 55
  • yy is greater than 3
  • zz is not equal to −1-1
  • aa is greater than 5 but less than or equal to 7
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Expressing an Inequality Algebraically

We can describe an inequality algebraically by using inequality symbols
Examples
  • x≤5x\le5 (xx is less than or equal to 55)
  • y>3y>3 (yy is greater than 3)
  • z≠−1z\neq -1 (zz is not equal to −1-1)
  • 5<a≤75<a\le7 (aa is greater than 5 but less than or equal to 7)
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Expressing an Inequality Graphically

We can describe the solutions to an inequality graphically by using a number line and following these rules

1. Highlight values on the number line that the variable can take on


2. Use arrows to show that a value goes on forever in one direction


3. Use an open cicle ∘\circ to represent not including the particular value and a closed circle ⋅\Large\cdot to represent including the particular value



A boundary point is the point that separates the values less than from the values greater than a specified value
  • If it's included in the solution to the inequality, we show it using a closed circle
  • If it's not included in the solution to the inequality, we show it using an open circle
checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

Practice: Represeting Inequalities

Sketch a number line to show each of the following inequalities.

a) x≤5x\le5

b) x≤2x\le2 and x>−1x>-1

c) −0.5≤x≤12-0.5\le x\le\dfrac{1}{2}
checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

Practice: Representing Inequalities

To ride a roller coaster at Canada's Wonderland, the rider must be at least 91cm tall. Represent this height restriction as an inequality.

a) verbally in words

b) algebraically using the appropriate inequality symbol

c) graphically on a number line.