Things You Must Know for Inequalities

Introduction to Inequalities

Inequalities are used to demonstrate relationships betwixt numbers or expressions.

Learning Objectives

Explain what inequalities stand for and how they are used

Primal Takeaways

Cardinal Points

• An inequality describes a relationship betwixt two different values.
• The notation [latex]a < b[/latex] means that [latex]a[/latex] is strictly smaller in size than [latex]b[/latex], while the annotation [latex]a > b[/latex] means that [latex]a[/latex] is strictly greater than [latex]b[/latex].
• The notion [latex]a \leq b[/latex] means that [latex]a[/latex] is less than or equal to [latex]b[/latex], while the notation [latex]a \geq b[/latex] ways that [latex]a[/latex] is greater than or equal to [latex]b[/latex].
• Inequalities are particularly useful for solving problems involving minimum or maximum possible values.

Key Terms

• number line: A visual representation of the ready of existent numbers as a series of points.
• inequality: A statement that of two quantities one is specifically less than or greater than another.

In mathematics, inequalities are used to compare the relative size of values. They can be used to compare integers, variables, and various other algebraic expressions. A description of unlike types of inequalities follows.

Strict Inequalities

A strict inequality is a relation that holds between two values when they are unlike. In the same way that equations use an equals sign, =, to show that two values are equal, inequalities use signs to show that 2 values are not equal and to describe their relationship. The strict inequality symbols are [latex]<[/latex] and [latex]>[/latex].

Strict inequalities differ from the note [latex]a \neq b[/latex], which means that a is not equal to [latex]b[/latex]. The [latex]\neq[/latex] symbol does not say that one value is greater than the other or even that they can be compared in size.

In the two types of strict inequalities, [latex]a[/latex] is not equal to [latex]b[/latex]. To compare the size of the values, there are two types of relations:

1. The note [latex]a < b[/latex] means that [latex]a[/latex] is less than [latex]b[/latex].
2. The annotation [latex]a > b[/latex] ways that [latex]a[/latex] is greater than [latex]b[/latex].

The meaning of these symbols can be easily remembered by noting that the "bigger" side of the inequality symbol (the open side) faces the larger number. The "smaller" side of the symbol (the point) faces the smaller number.

The above relations can exist demonstrated on a number line. Call back that the values on a number line increase every bit you lot move to the correct. The following therefore represents the relation [latex]a[/latex] is less than [latex]b[/latex]:

[latex]a < b[/latex]

[latex]a[/latex] is to the left of [latex]b[/latex] on this number line.

and the post-obit demonstrates [latex]a[/latex] being greater than [latex]b[/latex]:

[latex]a > b[/latex]

[latex]a[/latex] is to the correct of [latex]b[/latex] on this number line.

In general, note that:

• [latex]a < b[/latex] is equivalent to [latex]b > a[/latex]; for example, [latex]seven < 11[/latex] is equivalent to [latex]xi> 7[/latex].
• [latex]a > b[/latex] is equivalent to [latex]b < a[/latex]; for example, [latex]6 < nine[/latex] is equivalent to [latex]9 > half-dozen[/latex].

Other Inequalities

In contrast to strict inequalities, at that place are two types of inequality relations that are not strict:

• The notation [latex]a \leq b[/latex] means that [latex]a[/latex] is less than or equal to [latex]b[/latex] (or, equivalently, "at about" [latex]b[/latex]).
• The annotation [latex]a \geq b[/latex] means that [latex]a[/latex] is greater than or equal to [latex]b[/latex] (or, equivalently, "at least" [latex]b[/latex]).

Inequalities with Variables

In addition to showing relationships betwixt integers, inequalities can exist used to show relationships betwixt variables and integers.

For example, consider [latex]x > v[/latex]. This would be read as "[latex]10[/latex] is greater than five″ and indicates that the unknown variable [latex]x[/latex] could be any value greater than 5, though not 5 itself. For a visualization of this, meet the number line below:

[latex]x > v[/latex]

Note that the circle above the number 5 is open, indicating that 5 is not included in possible values of [latex]ten[/latex].

For another example, consider [latex]x \leq 3[/latex]. This would be read as "[latex]x[/latex] is less than or equal to 3″ and indicates that the unknown variable [latex]x[/latex] could be 3 or any value less than 3. For a visualization of this, see the number line below:

[latex]ten \leq three[/latex]

Note that the circle above the number 3 is filled, indicating that 3 is included in possible values of [latex]x[/latex].

Inequalities are demonstrated by coloring in an pointer over the appropriate range of the number line to point the possible values of [latex]x[/latex]. Note that an open circle is used if the inequality is strict (i.e., for inequalities using [latex]>[/latex] or [latex]<[/latex]), and a filled circumvolve is used if the inequality is not strict (i.e., for inequalities using [latex]\geq[/latex] or [latex]\leq[/latex]).

Solving Problems with Inequalities

Recall that equations tin can be used to demonstrate the equality of math expressions involving diverse operations (for case: [latex]x + 5 = 9[/latex]). Too, inequalities tin be used to demonstrate relationships between different expressions.

For instance, consider the following inequalities:

• [latex]10 - seven > 12[/latex]
• [latex]2x + 4 \leq 25[/latex]
• [latex]2x < y - 3[/latex]

Each of these represents the relationship between 2 different expressions.

One useful application of inequalities such as these is in problems that involve maximum or minimum values.

Example 1

Jared has a gunkhole with a maximum weight limit of two,500 pounds. He wants to accept every bit many of his friends as possible onto the boat, and he guesses that he and his friends weigh an average of 160 pounds. How many people can ride his boat at once?

This trouble can be modeled with the following inequality:

[latex]160n \leq 2500[/latex]

where [latex]n[/latex] is the number of people Jared tin can take on the boat. To run into why this is so, consider the left side of the inequality. It represents the total weight of [latex]n[/latex] people weighing 160 pounds each. The inequality states that the total weight of Jared and his friends should be less than or equal to the maximum weight of 2,500, which is the boat'south weight limit.

In that location are steps that can be followed to solve an inequality such as this one. For now, information technology is important simply to understand the meaning of such statements and cases in which they might exist applicable.

Rules for Solving Inequalities

Arithmetics operations tin be used to solve inequalities for all possible values of a variable.

Learning Objectives

Solve inequalities using the rules for operating on them

Key Takeaways

Key Points

• When you lot're performing algebraic operations on inequalities, it is of import to perform the same operation on both sides in social club to preserve the truth of the statement.
• If both sides of an inequality are multiplied or divided by the aforementioned positive value, the resulting inequality is truthful.
• If both sides are multiplied or divided by the same negative value, the direction of the inequality changes.
• Inequalities involving variables can exist solved to yield all possible values of the variable that make the statement true.

Key Terms

• inequality: A argument that of two quantities one is specifically less than or greater than another.

Operations on Inequalities

When you're performing algebraic operations on inequalities, it is of import to conduct precisely the same operation on both sides in club to preserve the truth of the statement.

Each arithmetic performance follows specific rules:

Improver and Subtraction

Any value [latex]c[/latex] may exist added to or subtracted from both sides of an inequality. That is to say, for any real numbers [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex]:

• If [latex]a \leq b[/latex], so [latex]a + c \leq b + c[/latex] and [latex]a - c \leq b - c[/latex].
• If [latex]a \geq b[/latex], so [latex]a + c \geq b + c[/latex] and [latex]a - c \geq b - c[/latex].

Every bit long as the aforementioned value is added or subtracted from both sides, the resulting inequality remains true.

For case, consider the following inequality:

[latex]12 < xv[/latex]

Let's apply the rules outlined above by subtracting 3 from both sides:

[latex]\brainstorm{marshal} 12 - 3 &< 15 - 3 \\ ix &< 12 \end{marshal}[/latex]

This argument is still true.

Multiplication and Division

The properties that deal with multiplication and division land that, for any real numbers, [latex]a[/latex], [latex]b[/latex], and non-cipher [latex]c[/latex]:

If [latex]c[/latex] is positive, then multiplying or dividing by [latex]c[/latex] does not modify the inequality:

• If [latex]a \geq b[/latex] and [latex]c >0[/latex], then [latex]ac \geq bc[/latex] and [latex]\dfrac{a}{c} \geq \dfrac{b}{c}[/latex].
• If [latex]a \leq b[/latex] and [latex]c > 0 [/latex], then [latex]ac \leq bc[/latex] and [latex]\dfrac{a}{c} \leq \dfrac{b}{c}[/latex].

If [latex]c[/latex] is negative, then multiplying or dividing by [latex]c[/latex] inverts the inequality:

• If [latex]a \geq b[/latex] and [latex]c <0 [/latex], then [latex]air conditioning \leq bc[/latex] and [latex]\dfrac{a}{c} \leq \dfrac{b}{c}[/latex].
• If [latex]a \leq b[/latex] and [latex]c < 0 [/latex], then [latex]ac \geq bc[/latex] and [latex]\dfrac{a}{c} \geq \dfrac{b}{c}[/latex].

Take note that multiplying or dividing an inequality by a negative number changes the direction of the inequality. In other words, a greater-than symbol becomes a less-than symbol, and vice versa.

To see these rules practical, consider the following inequality:

[latex]v > -three[/latex]

Multiplying both sides by 3 yields:

[latex]\begin{align} 5 (3) &> -three (3) \\ xv &> -nine \end{align}[/latex]

We see that this is a true statement, because xv is greater than 9.

At present, multiply the same inequality by -3 (call up to change the management of the symbol considering we're multiplying by a negative number):

[latex]\begin{align} 5 (-iii) &< -3 (-3) \\ -15 &< 9 \stop{marshal}[/latex]

This statement also holds true. This demonstrates how crucial information technology is to change the direction of the greater-than or less-than symbol when multiplying or dividing past a negative number.

Solving Inequalities

Solving an inequality that includes a variable gives all of the possible values that the variable can take that brand the inequality truthful. To solve an inequality ways to transform information technology such that a variable is on ane side of the symbol and a number or expression on the other side. Often, multiple operations are often required to transform an inequality in this way.

Addition and Subtraction

To see how the rules of improver and subtraction utilize to solving inequalities, consider the following:

[latex]x - 8 \leq 17[/latex]

First, isolate [latex]x[/latex]:

[latex]\begin{align} x - 8 + viii &\leq 17 + eight \\ x &\leq 25 \finish{align}[/latex]

Therefore, [latex]x \leq 25[/latex] is the solution of [latex]10 - 8 \leq 17[/latex]. In other words, [latex]x - viii \leq 17[/latex] is truthful for any value of [latex]ten[/latex] that is less than or equal to 25.

Multiplication and Division

To meet how the rules for multiplication and division use, consider the following inequality:

[latex]2x > 8[/latex]

Dividing both sides past 2 yields:

[latex]\begin{align} \dfrac{2x}{2} &> \dfrac{8}{2} \\ x &> \dfrac{eight}{2} \\ x &> 4 \end{align}[/latex]

The argument [latex]x > four[/latex] is therefore the solution to [latex]2x > viii[/latex]. In other words, [latex]2x > eight[/latex] is true for whatsoever value of [latex]x[/latex] greater than four.

Now, consider another inequality:

[latex]-\dfrac{y}{iii} \leq 7[/latex]

Because of the negative sign involved, we must multiply by a negative number to solve for [latex]y[/latex]. This ways that we must also change the direction of the symbol:

[latex]\begin{align} \displaystyle -iii \left( -\frac{y}{3} \right) &\geq -3 (vii)\\ y &\geq -3 (seven) \\ y &\geq -21 \end{align}[/latex]

Therefore, the solution to [latex]-\frac{y}{three} \leq 7[/latex] is [latex]y \geq -21[/latex]. The given argument is therefore truthful for any value of [latex]y[/latex] greater than or equal to [latex]-21[/latex].

Case

Solve the following inequality:

[latex]3y - 17 \geq xix[/latex]

Starting time, add 17 to both sides:

[latex]\begin{align} 3y - 17 + 17 &\geq 19 + 17 \\ 3y &\geq 36 \end{align}[/latex]

Adjacent, dissever both sides by 3:

[latex]\brainstorm{marshal} \dfrac{3y}{3} &\geq \dfrac{36}{iii} \\ y &\geq \dfrac{36}{3} \\ y &\geq 12 \finish{align}[/latex]

Special Considerations

Note that information technology would go problematic if nosotros tried to multiply or dissever both sides of an inequality by an unknown variable. If any variable [latex]x[/latex] is unknown, nosotros cannot identify whether information technology has a positive or negative value. Considering the rules for multiplying or dividing positive and negative numbers differ, we cannot follow this same rule when multiplying or dividing inequalities by variables. Variables tin can, however, be added or subtracted from both sides of an inequality.

Compound Inequalities

A chemical compound inequality involves 3 expressions, not two, but can as well exist solved to notice the possible values for a variable.

Learning Objectives

Solve a compound inequality by balancing all iii components of the inequality

Key Takeaways

Key Points

• A compound inequality is of the post-obit form: [latex]a < x < b[/latex].
• There are two statements in a compound inequality. The first statement is [latex]a < x[/latex]. The next argument is [latex]x < b[/latex]. When we read this statement, nosotros say "[latex]a[/latex] is less than [latex]x[/latex], and [latex]x[/latex] is less than [latex]b[/latex]."
• An instance of a compound inequality is: [latex]4 < 10 < 9[/latex]. In other words, [latex]x[/latex] is some number strictly between four and 9.
• A chemical compound inequality may contain an expression, such as [latex]1 < x - half-dozen < 8[/latex]; such inequalities can be solved for all possible values of [latex]x[/latex].

Central Terms

• chemical compound inequality: An inequality that is fabricated upward of ii other inequalities, in the form [latex]a < ten < b[/latex].
• inequality: A statement that of two quantities 1 is specifically less than or greater than another.

Defining Compound Inequalities

A compound inequality is of the post-obit class:

[latex]a < ten < b[/latex]

There are actually 2 statements here. The first argument is [latex]a < x[/latex]. The side by side statement is [latex]x < b[/latex]. This statement is therefore read every bit "[latex]a[/latex] is less than [latex]x[/latex], and [latex]ten[/latex] is less than [latex]b[/latex]."

The compound inequality [latex]a < ten < b[/latex] indicates "betweenness"—the number [latex]ten[/latex] is between the numbers [latex]a[/latex] and [latex]b[/latex]. Without irresolute the meaning, the statement [latex]a<x[/latex] tin can also be read every bit [latex]ten>a[/latex]. Therefore, the form [latex]a < x < b[/latex] tin can also be read as "[latex]ten[/latex] is greater than [latex]a[/latex], and at the aforementioned time is less than [latex]b[/latex]."

Consider [latex]4 < x < 9[/latex]. This states that [latex]x[/latex] is some number strictly between 4 and 9. For a visualization of this inequality, refer to the number line below. The numbers 4 and 9 are non included, then we place open up circles on these points.

[latex]iv < x < nine[/latex]

The above inequality on the number line.

Similarly, consider [latex]-2 < z < 0[/latex]. In this instance, [latex]z[/latex] is some number strictly between -2 and 0. Over again, considering the numbers -2 and 0 are not included, we identify open up circles on those points.

[latex]-ii < x < 0[/latex]

The above inequality on the number line.

[latex][/latex]Solving Compound Inequalities

At present consider [latex]1 < x + 6 < viii[/latex]. The expression [latex]10 + 6[/latex] represents some number strictly between 1 and 8. However, the meaning of this is difficult to visualize—what does information technology mean to say that an expression, rather than a number, lies between ii points? Non to worry—we tin can still find all possible values of not but the expression, but the variable [latex]x[/latex] itself.

The statement [latex]1 < x + 6 < 8[/latex] says that the quantity [latex]x + 6[/latex] is betwixt i and viii, a argument that volition be true for merely sure values of [latex]x[/latex].

To solve for possible values of [latex]x[/latex], we demand to get [latex]x[/latex] by itself:

[latex]1 - half-dozen < 10 + 6 - half-dozen < 8 - 6[/latex]

[latex]-5 < x < two[/latex]

Therefore, we find that if [latex]x[/latex] is whatever number strictly between -v and 2, the statement [latex]i < x + half-dozen < 8[/latex] will be true.

Instance ane

Solve [latex]-three < \dfrac{-2x-seven}{5} < 7[/latex].

Multiply each part to remove the denominator from the heart expression:

[latex]-three\cdot (5) < \dfrac{-2x-vii}{v} \cdot (v) < seven \cdot (5)[/latex]

[latex]-xv < -2x-7 < 35[/latex]

Isolate [latex]x[/latex] in the middle of the inequality:

[latex]- 15 + 7 < -2x -seven + 7 < 35 + 7[/latex]

[latex]- 8 < -2x < 42[/latex]

Now divide each part by -2 (and remember to change the management of the inequality symbol!):

[latex]\displaystyle \frac{-eight}{-2} > \frac{-2x}{-2} > \frac{42}{-2}[/latex]

[latex]4 > 10 > -21 [/latex]

Finally, it is customary (though not necessary) to write the inequality so that the inequality arrows bespeak to the left (i.e., so that the numbers proceed from smallest to largest):

[latex]-21 < x < 4[/latex]

Inequalities with Accented Value

Inequalities with absolute values can be solved by thinking well-nigh accented value as a number'due south distance from 0 on the number line.

Learning Objectives

Solve inequalities with absolute value

Key Takeaways

Key Points

• Problems involving absolute values and inequalities can exist approached in at least two ways: through trial and error, or by thinking of absolute value as representing distance from 0 then finding the values that satisfy that condition.
• When solving inequalities that involve an an absolute value within a larger expression (for example, [latex]\left|2x\right|+ 3>8[/latex]), it is necessary to algebraically isolate the absolute value and and so algebraically solve for the variable.

Fundamental Terms

• absolute value: The magnitude of a existent number without regard to its sign; formally, -1 times a number if the number is negative, and a number unmodified if it is zero or positive.
• inequality: A statement that of two quantities ane is specifically less than or greater than another.
• number line: A line that graphically represents the real numbers as a serial of points whose distance from an origin is proportional to their value.

Consider the following inequality that includes an absolute value:

[latex]|x| < 10[/latex]

Knowing that the solution to [latex] \left|ten\right|=10[/latex] is [latex]ten = ± 10[/latex], many students reply this question [latex]x < ± ten[/latex]. Nonetheless, this is wrong.

Hither are two different, just both perfectly right, ways to look at this problem.

Trial and Error

What numbers work? That is to say, for what numbers is [latex] \left|ten\right| < x[/latex] a true statement? Permit'south test some out.

4 works. -four does besides. 13 doesn't work. How virtually -13? No: If [latex]x = -13[/latex], then [latex] \left|x\right| = 13[/latex], which is not less than ten.

By playing with numbers in this way, you should exist able to convince yourself that the numbers that work must exist somewhere between -x and 10. This is one way to arroyo finding the answer.

Absolute Value as Distance

The other way is to recollect of absolute value as representing distance from 0. [latex] \left|five\correct|[/latex]and [latex] \left|-5\correct|[/latex] are both 5 considering both numbers are 5 abroad from 0.

In this case, [latex] \left|10\right| < x[/latex] means "the distance between [latex]x[/latex] and 0 is less than 10." In other words, you are within 10 units of naught in either management. Once once again, we conclude that the answer must be between -10 and x.

This answer tin can exist visualized on the number line as shown below, in which all numbers whose absolute value is less than 10 are highlighted.

Solution to [latex] \left|x\right| < ten[/latex]: All numbers whose absolute value is less than 10.

It is not necessary to utilise both of these methods; use whichever method is easier for you to understand.

Solving Inequalities with Absolute Value

More than complicated absolute value problems should exist approached in the same way equally equations with absolute values: algebraically isolate the accented value, and then algebraically solve for [latex]x[/latex].

For example, consider the following inequality:

[latex] \left|2x\right| + three>viii[/latex]

It is hard to immediately visualize the meaning of this absolute value, let alone the value of [latex]x[/latex] itself. It is necessary to first isolate the inequality:

[latex]\begin{marshal} \left|2x\right| + iii - 3 &> 8 - 3 \\ \left|2x\right| &> eight \end{align}[/latex]

Now think about the number line. In those terms, this statement means that the expression [latex]2x[/latex] must exist more than 8 places away from 0. Therefore, it must be either greater than 8 or less than -8. Expressing this with inequalities, we have:

[latex]2x>viii[/latex] or [latex]2x < -8[/latex]

We at present take 2 separate inequalities. If each one is separately solved for [latex]x[/latex], nosotros volition see the full range of possible values of [latex]x[/latex]. Consider them independently. Start:

[latex]\begin{align} 2x &>8 \\ \dfrac{2x}{ii} &> \dfrac{eight}{2} \\ x &> iv \end{align} [/latex]

Second:

[latex]\brainstorm{align} 2x &< -8 \\ \dfrac{2x}{2} &< \dfrac{-8}{2} \\ x &< -4 \end{align}[/latex]

Nosotros at present have two ranges of solutions to the original absolute value inequality:

[latex]x > 4[/latex] and [latex]10 < -iv[/latex]

This can too be visually displayed on a number line:

Solution to [latex]\left|2x\right| + 3>8[/latex]: The solution is any value of [latex]ten[/latex] less than -iv or greater than 4.

Example

Solve the following inequality:

[latex] \left|x-two\right| + 10 > seven[/latex]

Start, algebraically isolate the absolute value:

[latex]\brainstorm{align} \left|ten-2\correct| + 10 - 10 &> 7 - 10 \\ \left|x-2\correct| &> - 3 \end{marshal}[/latex]

At present retrieve: the absolute value of the expression is greater than –three. What could the expression exist equal to? 2 works. –2 also works. And 0. And seven. And –ten. Absolute values are always positive, so the absolute value of anything is greater than –3! All numbers therefore work.

bennettdiell1982.blogspot.com

Source: https://courses.lumenlearning.com/boundless-algebra/chapter/inequalities/