Writing and Graphing One-Variable Inequalities

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Equations are mathematical statements relating two equal quantities. There are also times when it's necessary to relate two quantities that are not equal. For this, inequalities are used.
Concept

Inequality

An inequality, similar to an equation, is a mathematical statement comparing two quantities. Some inequalities indicate that the two quantities are not necessarily equal, while others are strictly never equal.

Symbol Meaning Strict / Non-strict
<\lt is less than Strict
\leq is less than or equal to Non-strict
>\gt is greater than Strict
\geq is greater than or equal to Non-strict
With an inequality, it's possible to compare any combination of two numbers, variables, numeric expressions, or algebraic expressions.
Concept

Solution Set of an Inequality

When an inequality contains an unknown variable, it's possible to solve the inequality. A solution of an inequality is any value of the variable that makes the inequality true. For instance, the inequality x+2<5 x + 2 < 5

has the solution x=1,x = 1, because replacing xx with 11 yields 3<5,3 < 5, a true statement. Notice that x=1x=1 is not the only value that solves x+2<5;x+2<5; x=0x=0 and x=2x=2 also work. In fact, most inequalities have an infinite number of solutions. The set of these solutions is called the solution set.
Exercise

Is r=2r = 2 an element of the solution set of the following inequality? 3r-1 \dfrac{3}{r} \leq \text{-} 1

Solution

If a value is a solution to an inequality, it is an element of the solution set. To determine if r=2r=2 is a solution, we can substitute the value into the inequality and evaluate. If the resulting inequality is true, the value is a solution; if it's false, it is not a solution.

3r?-1\dfrac{3}{r} \stackrel{?}{\leq} \text{-} 1
32?-1\dfrac{3}{{\color{#0000FF}{2}}} \stackrel{?}{\leq} \text{-} 1
1.5-11.5 \nleq \text{-} 1
The inequality 1.5-1,1.5 \leq \text{-} 1, read as 1.5``1.5 is less than or equal to -1,"\text{-} 1," is a false statement. Thus, r=2r = 2 is not a solution of the inequality, nor is it an element of the solution set.
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Concept

Graphing an Inequality on a Number Line

The graph of a one-variable inequality is a visual representation of the inequality's solution set, which can be drawn on a number line in three steps:

  1. Identify if the inequality is strict.
  2. Draw the boundary point;
    • use an open point ()(\circ) for strict or
    • use a closed point ()(\bullet) for non-strict.
  3. Shade the rest of the solution set.

An arrow in either direction indicates that all numbers in that direction are part of the solution set.

x0x \ge 0

x2x \leq 2

x>-3x > \text{-} 3

The inequality x0x \geq 0 is graphed with a closed circle at x=0x = 0 and an arrow pointing to the right. Since 00 solves the inequality, the circle is closed. Additionally, the direction of the arrow points toward numbers greater than 00, since the inequality symbol is .\geq.
Exercise

Graph the inequality x<9x < 9 on a number line.

Solution

The inequality reads x``x is less than 9,"9," which means that x=9x = 9 is not a solution to the inequality, but every value of xx less than 99 is. Thus, our graph must have an open circle at x=9.x = 9.

Every value of xx less than 99 also has to be included in the graph. Since smaller numbers lie to the left on the number line, this is graphed as an arrow pointing to the left.

We have now fully graphed the inequality x<9.x < 9.

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Exercise

Write the inequality shown by the graph.

Solution

To begin, let's use the variable xx in the inequality. From the graph, we can see a closed circle at -3.\text{-} 3. Therefore, x=-3x = \text{-} 3 is a solution of the inequality. The arrow pointing toward the right indicates that all values greater than 33 are also part of the solution set. Thus, the inequality would read x``x is greater than or equal to -3,"\text{-} 3," which is expressed algebraically as x-3. x \geq \text{-} 3.

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Exercises

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