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1. Equations & Inequalities in One Variable
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United States of AmericaTennessee
Integrated Math 1

1.06 Multi-step inequalities

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Lesson

Concept summary

Some mathematical relations compare two non-equivalent expressions - these are known as inequalities.

We can solve inequalities by using various properties to isolate the variable, in a similar way to solving equations:

Asymmetric property of inequality

If a > b, then b<a

Transitive property of inequality

If a>b and b>c, then a>c

Addition property of inequality

If a>b, then a+c>b+c

Subtraction property of inequality

If a>b, then a-c>b-c

Multiplication property of inequality

If a>b and c>0, then a \cdot c > b \cdot c

If a>b and c<0, then a \cdot c < b \cdot c

Division property of inequality

If a>b and c>0, then a \div c > b \div c.

If a>b and c<0, then a \div c < b \div c.

When multiplying or dividing an inequality by a negative value the inequality symbol is reversed.

Linear inequality

An inequality that contains a variable term with an exponent of 1, and no variable terms with exponents other than 1

Example:

3x - 5 \geq 4

Solving an inequality using the above properties of inequalities results in a solution set.

Solution set

The set of all values that make the inequality or equation true

Example:

3 \leq x

We can represent solutions to inequalities algebraically, by using numbers, letters, and/or symbols, or graphically, by using a coordinate plane or number line.

An algebraic solution to an inequality can be represented as an inequality, such as 3\leq x, or in other ways, including interval notation and set-builder notation.

Interval

A set of numbers that lie between two values

Interval notation

A way to represent a solution set or interval as a pair of numbers using a combination of square brackets and parentheses.

Example:

Inequality solution

3 \leq x

Interval notation

\left[ 3 , \infty \right)

We use square brackets if the endpoint is included and parentheses if the endpoint is not included. We always use parenthese for infinity. We can join two sets together using the union symbol \cup. We may see x \in which says "x is in".

Set-builder notation

A shorthand used to write sets. The set \left\{x \in \Reals \vert x > 0\right\} is read as "the set of all real values of x such that x is greater than zero". Sometimes a colon : is used instead of a vertical line \vert.

Example:

Inequality solution

3 \leq x

Set-builder notation

\left\{x \in \Reals \vert x \geq 3 \right\}

-5-4-3-2-1012345
  • Inequality notation: -3 \leq x <4
  • Interval notation: \left[-3,4\right)
  • Set-builder notation: \left\{x \in \Reals \vert -3 \leq x <4 \right\}
-5-4-3-2-1012345
  • Inequality notation: x\geq-3
  • Interval notation: \left[-3,\infty\right)
  • Set-builder notation: \left\{x \in \Reals \vert x\geq-3 \right\}

Two inequalities that have the same set of solutions are called equivalent inequalities.

Worked examples

Example 1

Consider the inequality \dfrac{-8-3x}{2} \leq 5.

a

Solve the inequality.

Approach

We want to isolate x on one side of the inequality and a number on the other.

Solution

\displaystyle \frac{-8-3x}{2}\displaystyle \leq\displaystyle {5}
\displaystyle -8-3x\displaystyle \leq\displaystyle 10Multiplication property of inequality
\displaystyle -3x\displaystyle \leq\displaystyle 18Addition property of inequality
\displaystyle x\displaystyle \geq\displaystyle -6Division property of inequality

Reflection

Solving an inequality is similar to solving an equation. However, we need to reverse the direction of the inequality when multiplying or dividing by a negative number.

b

Plot the inequality on a number line.

Solution

Plot the solution set of the inequality x \geq -6. Note that since we include -6 the point should be filled.

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

Reflection

What if the solution was x \gt -6?

End points included in the interval are filled points.

End points not included in the interval are unfilled points.

Example 2

Oprah charges \$ 37.72 to style hair, as well as \$ 6 per foil. Pauline would like a style and foils, but has no more than \$ 95.86 to spend.

a

Write an inequality that represents the number of foils Pauline could get.

Approach

Pauline would like a style which is \$37.72 and some foils at \$6 each. If the number of foils Pauline can get is N, then we can write an expression that represents the cost to style hair as 6N + \$37.72.

Pauline has no more than \$ 95.86 to spend. Remember that "no more than" means "less than or equal to."

Solution

6N +37.72 \leq 95.86

b

Write the solution set to the inequality.

Approach

Solve the inequality and then write the solution set.

Solution

\displaystyle 37.72 + 6N\displaystyle \leq\displaystyle 95.86
\displaystyle 6N\displaystyle \leq\displaystyle 58.14Subtraction property of inequality
\displaystyle N\displaystyle \leq\displaystyle 9.69Division property of inequality

Inequality notation: N\leq 9.69

Reflection

We could also write this in interval notation as \left(-\infty, 9.69 \right].

c

Determine the solution set in the context of the question.

Approach

We need to work out what the solution means in context.

Solution

Pauline cannot get part of a foil, so she can get a maximum of 9 foils and a minimum of 0 foils. In context, the solution set contains the following numbers 0,1,2,3,4,5,6,7,8, and 9 because Pauline can get any number between 0 and 9 foils.

Reflection

Sometimes the context restricts the set of solutions.

Outcomes

M1.N.Q.A.1

Use units as a way to understand real-world problems.*

M1.N.Q.A.1.C

Define and justify appropriate quantities within a context for the purpose of modeling.*

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.CED.A.3

Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.*

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.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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