FoundationsConcept reference

Inequalities

A solution of a scalar inequality is any value that makes its ordered comparison true.

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On this page 7 sections

Overview
An inequality describes a range of allowed values
The bar still matters after solving
Check your understanding
Key takeaway
Sources & further reading
Concept connections

Figure 1Solution of x+2<7: the ray x<5 extends left from an open endpoint at 5. The endpoint is excluded.

01An inequality describes a range of allowed values#

An inequality describes a range of allowed values. The sign $<$ means less than, $>$ greater than. Adding a bar, as in $\le$ or $\ge$ , allows equality too.

The expression on the left must be below 7.

x+2<7

Starting from $x+2<7$ , subtract $2$ from both sides. Moving both values by the same amount keeps their order.

Solve the one-step inequality.

x<5

Multiplying or dividing by a positive number also keeps the order. A negative multiplier reverses it: $2<5$ , but $-2>-5$ .

Divide both sides of $-2x<6$ by −2 and reverse the sign.

x>-3

Check your reasoning

Solve $-3x\ge12$ .

A $x\ge-4$
B $x\le-4$
C $x\le4$

Show answer and explanation

$x\le-4$

Divide by $-3$ , reversing $\ge$ to $\le$ . The endpoint is $12/(-3)=-4$ .

02The bar still matters after solving#

The bar still matters after solving. For $x\le4$ , the value $4$ is allowed; for $x<4$ , it is not. Dividing by a negative reverses direction but does not remove equality.

For the condition $-2x<6$ , test $x=0$ .

-2(0)<6\quad\text{is true}

Check your reasoning

A display allows $x+3\le11$ . What is the largest allowed value of $x$ ?

Show answer and explanation

Subtract $3$ : $x\le8$ . Equality is allowed, so $8$ is included.

03Check your understanding#

Check your reasoning

A sensor rule is $-5x>10$ . Which range satisfies it?

A $x>-2$
B $x<2$
C $x<-2$

Show answer and explanation

$x<-2$

Dividing by $-5$ reverses the inequality and gives $x<-2$ .

Key takeaway

Undo the operation on both sides. Reverse the inequality only when multiplying or dividing by a negative number; preserve whether equality is allowed.

Solve a one-step scalar inequality

Sources & further reading

[1]
OpenStax Elementary Algebra 2e, 2.7 ↗OpenStax · Book

01An inequality describes a range of allowed values#

02The bar still matters after solving#

03Check your understanding#

Undo the operation on both sides. Reverse the inequality only when multiplying or dividing by a negative number; preserve whether equality is allowed.

Sources & further reading

Reference this concept