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1. Equations & Inequalities in One Variable
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Integrated Math 1

1.07 Compound inequalities

Lesson
Practice
Lesson

Concept summary

A compound inequality is a conjuction of two or more inequalities. The set of solutions for a compound inequality are the values which make all of the inequalities true.

-5-4-3-2-1012345
  • We use "and" to indicate that a value must satisfy both inequalities in order to be in the solution set. For example:x \lt 3 \text{ and } x \geq -2We can also write this compound inequality more simply as -2\leq x <3
-5-4-3-2-1012345
  • We use "or" to indicate that a value need only satisfy at least one inequality in order to be in the solution set. For example: x \gt 3 \text{ or } x \leq -2

Worked examples

Example 1

Write a compound inequality to represent the solution set shown on the number line.

-5-4-3-2-1012345

Approach

We want to describe the compound inequality then write the solution to the compound inequality algebraically.

Description: x is less than -1 or x is greater than or equal to 2

Solution

Compound inequality: x < -1 or x \geq 2

Example 2

Find the solution set of the compound inequality x \geq -5 and x < 3 using set-builder notation and plot the solution on a number line.

Approach

  1. Plot x \geq -5 on a number line.

  2. Plot x < 3 on a number line.

  3. Find the solution of the compound inequality by comparing the number lines.

  4. Represent the solution graphically.

  5. Represent the solution using set-builder notation.

Solution

Plotting x \geq -5:

-6-4-20246

The minimum value, -5, is included in the interval and is represented by a filled circle.

Plotting x< 3:

-6-4-20246

The maximum value, 3, is not included in the interval and is represented by a unfilled circle.

Finding the solution to the compound inequality:

We want the solution to both x \geq -5 and x \lt 3. The solution to the compound inequality will be the values greater than or equal to -5 and less than 3.

Solution represented graphically on a number line:

-10-9-8-7-6-5-4-3-2-1012345678910

Solution represented algebraically using set-builder notation: \left\{x\in\Reals\vert -5\leq x<3\right\}

Reflection

We can check our solution(s) are correct by taking a number in our solution set and seeing if it works in both inequalities.

For example, we can see that 0 falls within the region shown on the number line, and \\-5\leq 0 < 3 as required.

Example 3

The formula for converting temperatures from Celsius to Fahrenheit is: F = \dfrac{9}{5} C+32

During a recent year, the average temperatures in Tampa, Florida ranged from 59 \degree to 95 \degree Fahrenheit.

Write a compound inequality to solve for the corresponding range of values of C, the temperature in Florida in degrees Celsius.

Approach

The degrees in Fahrenheit must be between 59 and 95 inclusive, so the minimum is 59 and the maximum is 95. The middle part of our compound inequality is going to be the formula for converting temperatures.

Once we have the compound inequality we can solve it.

Solution

\displaystyle 59\displaystyle \leq\displaystyle \frac{9}{5} C+32 \leq 95
\displaystyle 27\displaystyle \leq\displaystyle \frac{9}{5}C \leq 63Subtraction property of inequality
\displaystyle 135\displaystyle \leq\displaystyle 9C \leq 315Multiplication property of inequality
\displaystyle 15\displaystyle \leq\displaystyle C \leq 35Division property of inequality

The average temperatures in Tampa ranged from 15 \degree to 35 \degree Celsius.

Reflection

When solving a compound inequality, whatever is done to one part of the inequality must be done to all parts.

Example 4

Consider the following pair of inequalities:\begin{aligned} \text{Inequality 1: }\, \, & 1.2x +0.2 \leq 11.6 \\ \text{Inequality 2: }\,\, & 6 - 3.5x \lt 48 \end{aligned}

a

State the solution to the compound inequality that is Inequality 1 OR Inequality 2.

Approach

We need to rearrange the inequalities to isolate x. To rearrange these inequalities, we will consider each case separately and use the properties of inequalities and inverse operations to find the solution.

Solution

Inequality 1:

\displaystyle 1.2x +0.2\displaystyle \leq\displaystyle 11.6
\displaystyle 1.2x\displaystyle \leq\displaystyle 11.4Subtraction property of inequality
\displaystyle x\displaystyle \leq\displaystyle 9.5Division property of inequality

Inequality 2:

\displaystyle 6-3.5x\displaystyle <\displaystyle 48
\displaystyle -3.5x\displaystyle <\displaystyle 42Subtraction property of inequality
\displaystyle x\displaystyle >\displaystyle 12Division property of inequality

The solution set is the compound inequality: x \leq 9.5 or x\gt 12

b

Plot the solution set on a number line.

Approach

  1. Plot x \leq 9.5 on a number line.

  2. Plot x \gt 12 on a number line.

  3. Find the solution of the compound inequality by comparing the number lines and considering the given condition.

  4. Represent the solution graphically.

Solution

Plotting x \leq 9.5:

56789101112

The maximum value, 9.5, is included in the interval and is represented by a filled circle.

Plotting x \gt 12:

9101112131415

The minimum value, 12, is not included in the interval and is represented by a unfilled circle.

Finding the solution to the compound inequality:

We want the solution to be either x \leq 9.5 or x \gt 12. As the compound inequality indicates "or", and we do not have overlapping conditions, we draw the two intervals on the same number line to represent the full solution set.

Solution represented graphically on a number line:

567891011121314151617

Reflection

What would the solution set look like if the condition was "and" rather than "or"?

For an "and" condition we require both inequalities hold. As there in no overlap between the two intervals, there are no values that satisfy both inequalities. In this case, there would be no solutions to the compound inequality.

c

Determine whether x=7.5 is a valid solution to the compound inequality.

Approach

Plot the solution set togther with the indicated point to check if the point lies inside or outside the solution set. Or test algebraically by substituting the value into either the original or rearranged inequalities and check if the resulting statement is true.

Solution

Graphically:

Consider the plot of the solution set together with the point at x=7.5:

567891011121314151617

The point x=7.5 lies on a section of the line that in the solution set, indicated by the green arrow, and therefore is a valid solution.

Algebraically:

Substituting x=7.5 into Inequality 1:

\displaystyle 1.2x +0.2\displaystyle \leq\displaystyle 11.6
\displaystyle 1.2\left(7.5\right)+0.2\displaystyle \leq\displaystyle 11.6Substitute x=7.5 into the inequality
\displaystyle 9+0.2\displaystyle \leq\displaystyle 11.6Evaluate the multiplication
\displaystyle 9.2\displaystyle \leq\displaystyle 11.6Evaluate the addition

As this statement is true, x=7.5 is a valid solution.

Reflection

As we had a compound equality with an "or" condition we can stop at checking that one of the inequalities hold. For an "and" condition we would need to check both inequalities hold.

Outcomes

M1.A.CED.A.1

Create equations and inequalities in one variable and use them to solve problems in a real-world context.*

M1.A.REI.B.2

Solve linear and absolute value equations and inequalities in one variable.

M1.A.REI.B.2.A

Solve linear equations and inequalities, including compound inequalities, in one variable. Represent solutions algebraically and graphically.

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

M1.MP8

Look for and express regularity in repeated reasoning.

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