Writing and Manipulating Inequalities
Learning Outcomes
- Use interval notation to express inequalities.
- Use properties of inequalities.
Inequality | Words | Interval Notation |
---|---|---|
[latex]{a}\lt{x}\lt{ b}[/latex] | all real numbers between a and b, not including a and b | [latex]\left(a,b\right)[/latex] |
[latex]{x}\gt{a}[/latex] | All real numbers greater than a, but not including a | [latex]\left(a,\infty \right)[/latex] |
[latex]{x}\lt{b}[/latex] | All real numbers less than b, but not including b | [latex]\left(-\infty ,b\right)[/latex] |
[latex]{x}\ge{a}[/latex] | All real numbers greater than a, including a | [latex]\left[a,\infty \right)[/latex] |
[latex]{x}\le{b}[/latex] | All real numbers less than b, including b | [latex]\left(-\infty ,b\right][/latex] |
[latex]{a}\le{x}\lt{ b}[/latex] | All real numbers between a and b, including a | [latex]\left[a,b\right)[/latex] |
[latex]{a}\lt{x}\le{ b}[/latex] | All real numbers between a and b, including b | [latex]\left(a,b\right][/latex] |
[latex]{a}\le{x}\le{ b}[/latex] | All real numbers between a and b, including a and b | [latex]\left[a,b\right][/latex] |
[latex]{x}\lt{a}\text{ or }{x}\gt{ b}[/latex] | All real numbers less than a or greater than b | [latex]\left(-\infty ,a\right)\cup \left(b,\infty \right)[/latex] |
All real numbers | All real numbers | [latex]\left(-\infty ,\infty \right)[/latex] |
Example: Using Interval Notation to Express an inequality
Use interval notation to indicate all real numbers greater than or equal to [latex]-2[/latex].Answer: Use a bracket on the left of [latex]-2[/latex] and parentheses after infinity: [latex]\left[-2,\infty \right)[/latex]. The bracket indicates that [latex]-2[/latex] is included in the set with all real numbers greater than [latex]-2[/latex] to infinity.
example: using interval notation to express an inequality
Describe the inequality [latex]x\ge 4[/latex] using interval notationAnswer: The solutions to [latex]x\ge 4[/latex] are represented as [latex]\left[4,\infty \right)[/latex]. Note the use of a bracket on the left because 4 is included in the solution set.
Try It
Use interval notation to indicate all real numbers between and including [latex]-3[/latex] and [latex]5[/latex].Answer: [latex-display]\left[-3,5\right][/latex-display]
[ohm_question]58-92604[/ohm_question]Example: Using Interval Notation to Express a compound inequality
Write the interval expressing all real numbers less than or equal to [latex]-1[/latex] or greater than or equal to [latex]1[/latex].Answer: We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at [latex]-\infty [/latex] and ends at [latex]-1[/latex], which is written as [latex]\left(-\infty ,-1\right][/latex]. The second interval must show all real numbers greater than or equal to [latex]1[/latex], which is written as [latex]\left[1,\infty \right)[/latex]. However, we want to combine these two sets. We accomplish this by inserting the union symbol, [latex]\cup [/latex], between the two intervals.
Try It
Express all real numbers less than [latex]-2[/latex] or greater than or equal to 3 in interval notation.Answer: [latex-display]\left(-\infty ,-2\right)\cup \left[3,\infty \right)[/latex-display]
try it
We are going to look at a line with endpoints along the x-axis.- First we will adjust the left endpoint to (-15,0), and the right endpoint to (5,0)
- Write an inequality that represents the line you created.
Answer: With endpoints (-15,0) and (5,0), the values for x on the line are between -15 and 5, so we can write [latex]-15<x<5[/latex]. We made it a strict inequality because the dots on the endpoints of the lines are open. Moving the left endpoint towards the right endpoint shortens the line. Then moving the right endpoint away from the left endpoint lengthens the line again.
think about it
In the previous examples you were given an inequality or a description of one with words and asked to draw the corresponding graph and write the interval. In this example you are given an interval and asked to write the inequality and draw the graph. Given [latex]\left(-\infty,10\right)[/latex], write the associated inequality and draw the graph. In the box below, write down whether you think it will be easier to draw the graph first or write the inequality first. [practice-area rows="1"][/practice-area]Answer: We will draw the graph first. The interval reads "all real numbers less than 10," so we will start by placing an open dot on 10 and drawing a line to the left with an arrow indicating the solution continues to negative infinity. To write the inequality, we will use < since the parentheses indicate that 10 is not included. [latex]x<10[/latex]
Using the Properties of Inequalities
recall solving multi-step equations
When solving inequalities, all the properties of equality and real numbers apply. We are permitted to add, subtract, multiply, or divide the same quantity to both sides of the inequality. Likewise, we may apply the distributive, commutative, and associative properties as desired to help isolate the variable. We may also distribute the LCD on both sides of an inequality to eliminate denominators. The only difference is that if we multiply or divide both sides by a negative quantity, we must reverse the direction of the inequality symbol.A General Note: Properties of Inequalities
[latex]\begin{array}{ll}\text{Addition Property}\hfill& \text{If }a< b,\text{ then }a+c< b+c.\hfill \\ \hfill & \hfill \\ \text{Multiplication Property}\hfill & \text{If }a< b\text{ and }c> 0,\text{ then }ac< bc.\hfill \\ \hfill & \text{If }a< b\text{ and }c< 0,\text{ then }ac> bc.\hfill \end{array}[/latex]
These properties also apply to [latex]a\le b[/latex], [latex]a>b[/latex], and [latex]a\ge b[/latex].Example: Demonstrating the Addition Property
Illustrate the addition property for inequalities by solving each of the following:- [latex]x - 15<4[/latex]
- [latex]6\ge x - 1[/latex]
- [latex]x+7>9[/latex]
Answer: The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality. 1. [latex-display]\begin{array}{ll}x - 15<4\hfill & \hfill \\ x - 15+15<4+15 \hfill & \text{Add 15 to both sides.}\hfill \\ x<19\hfill & \hfill \end{array}[/latex-display] 2. [latex-display]\begin{array}{ll}6\ge x - 1\hfill & \hfill \\ 6+1\ge x - 1+1\hfill & \text{Add 1 to both sides}.\hfill \\ 7\ge x\hfill & \hfill \end{array}[/latex-display] 3. [latex-display]\begin{array}{ll}x+7>9\hfill & \hfill \\ x+7 - 7>9 - 7\hfill & \text{Subtract 7 from both sides}.\hfill \\ x>2\hfill & \hfill \end{array}[/latex-display]
Try It
Solve [latex]3x - 2<1[/latex].Answer: [latex-display]x<1[/latex-display]
[ohm_question]92605[/ohm_question]Example: Demonstrating the Multiplication Property
Illustrate the multiplication property for inequalities by solving each of the following:- [latex]3x<6[/latex]
- [latex]-2x - 1\ge 5[/latex]
- [latex]5-x>10[/latex]
Answer: 1. [latex-display]\begin{array}{l}3x<6\hfill \\ \frac{1}{3}\left(3x\right)<\left(6\right)\frac{1}{3}\hfill \\ x<2\hfill \end{array}[/latex-display] 2. [latex-display]\begin{array}{ll}-2x - 1\ge 5\hfill & \hfill \\ -2x\ge 6\hfill & \hfill \\ \left(-\frac{1}{2}\right)\left(-2x\right)\ge \left(6\right)\left(-\frac{1}{2}\right)\hfill & \text{Multiply by }-\frac{1}{2}.\hfill \\ x\le -3\hfill & \text{Reverse the inequality}.\hfill \end{array}[/latex-display] 3. [latex-display]\begin{array}{ll}5-x>10\hfill & \hfill \\ -x>5\hfill & \hfill \\ \left(-1\right)\left(-x\right)>\left(5\right)\left(-1\right)\hfill & \text{Multiply by }-1.\hfill \\ x<-5\hfill & \text{Reverse the inequality}.\hfill \end{array}[/latex-display]
Try It
Solve [latex]4x+7\ge 2x - 3[/latex].Answer: [latex]x\ge -5[/latex]
[ohm_question]92606[/ohm_question]Solving Inequalities in One Variable Algebraically
As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.Example: Solving an Inequality Algebraically
Solve the inequality: [latex]13 - 7x\ge 10x - 4[/latex].Answer: Solving this inequality is similar to solving an equation up until the last step.
Try It
Solve the inequality and write the answer using interval notation: [latex]-x+4<\frac{1}{2}x+1[/latex].Answer: [latex]\left(2,\infty \right)[/latex]
[ohm_question]92607[/ohm_question]Example: Solving an Inequality with Fractions
Solve the following inequality and write the answer in interval notation: [latex]-\frac{3}{4}x\ge -\frac{5}{8}+\frac{2}{3}x[/latex].Answer: We begin solving in the same way we do when solving an equation.
Try It
Solve the inequality and write the answer in interval notation: [latex]-\frac{5}{6}x\le \frac{3}{4}+\frac{8}{3}x[/latex].Answer: [latex-display]\left[-\frac{3}{14},\infty \right)[/latex-display]
[ohm_question]72891[/ohm_question]Licenses & Attributions
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- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
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- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
- Question ID 92604, 92605, 92606, 92607. Authored by: Michael Jenck. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
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- Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Left Side). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Right Side). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Specific attribution
- College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.