Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
128
Inequalities in One Variable
Dust off your equation-solving skills from Chapter 4
7.1 Identify the five most commonly used inequality symbols.
The five most used algebraic inequality symbols are: <, less than; >, greater
than; ≤, less than or equal to; ≥, greater than or equal to; and ≠, not equal to.
When possible, use <, ≤, >, and ≥ instead of ≠, because they communicate more
information.
7.2 Of the inequality symbols <, ≤, >, and ≥, which correctly complete the
following statement?
2 ______ 7
Two is fewer than seven, so the less than symbol correctly completes the
statement: 2 < 7. Two is also less than or equal to seven. For the statement 2 ≤ 7 to
be true, exactly one of the following conditions must be met: two must either be
less than seven (it is), or it must be equal to seven (it is not).
7.3 Of the inequality symbols <, ≤, >, and ≥, which correctly complete the
following statement?
–4 ______ –4
As stated in Problem 7.2, a statement containing the ≤ symbol is true in one of
two cases: if the left quantity is less than the right quantity or both quantities
are equal. Here, the left and right sides of the inequality are equal, so the
statement –4 ≤ –4 is true. Similarly, the statement –4 ≥ –4 is true.
7.4 Is the following statement true or false? Explain your answer.
–15 > –12
The statement –15 > –12 is false. The more negative a number, the less that
number is considered. Therefore, –15 is less than –12 because –15 is more
negative than –12.
7.5 Solve the inequality x – 3 > 11 for x.
To solve a linear inequality in one variable, isolate that variable left of the
inequality sign, much like you would solve a linear equation by isolating the
variable left of the equal sign. Here, isolate x by adding 3 to both sides of
the inequality.
Don’t say, “I
am not the same
height as my cousin,”
when you can say, “I
am shorter than my
cousin.” The second
statement doesn’t just
state that you have
different heights—
it also explains
why.
There’s no way
you can meet BOTH
conditions. No number
is both equal to
itself AND less than
itself.
If –4 ≥ –4
is true, then
either the left
side is bigger than
the right side (it’s not)
or both sides are
equal (they are).
Think of
a number
line. If you plot
–12 and –15, –15
is farther to the
left on the number
line. The farther left
you go, the “less” the
number. The farther
right you go, the
“greater” the
number.
Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
129
The solution to the inequality is x > 14, so substituting any real number greater
than 14 into the inequality x – 3 > 11 produces a true statement.
7.6 Solve the inequality 4x + 1 ≤ –11 for x.
Isolate the x-term on the left side of the inequality by subtracting 1 from both
sides of the equation.
Divide both sides of the inequality by 4 to solve for x.
7.7 Solve the inequality 7x – 2(x + 1) ≤ 18.
Expand and simplify the left side of the inequality.
Isolate 5x on the left side of the inequality and then divide both sides by 5 to
eliminate the coefficient.
7.8 Identify the four most common reasons an inequality sign must be reversed.
The four most common reasons an inequality symbol must be reversed are:
multiplying both sides of an inequality by a negative number, dividing both
sides of an inequality by a negative number, exchanging the sides of an
inequality, and taking the reciprocal of both sides of an inequality.
Not including 14
Reversing
an inequality
symbol means
changing the
direction it points.
That means <
becomes > (and
vice versa) and ≤
becomes ≥ (and
vice versa).
If you swap
the sides of an
inequality, you
have to reverse the
inequality symbol.
For example, 3 > x
becomes x < 3.
Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
130
7.9 Solve the inequality for x.
To isolate x, multiply both sides of the inequality by , the reciprocal of its
coefficient.
It is customary to position x left of the inequality symbol. Exchange the sides of
the inequality and, as noted in Problem 7.8, reverse the inequality symbol.
7.10 Solve the inequality for x.
Subtract from, and add 1 to, both sides of the inequality to separate the
x-terms and constants on opposite sides of the inequality symbol.
Combine like terms, using the least common denominator 10 to add the
fractions.
To eliminate the coefficient of x, multiply by . As indicated in Problem
7.8, multiplying both sides of an inequality by a negative number requires the
reversal of the inequality sign.
If x is on the
left side of the
inequality, then it’s
slightly easier
to graph—the
symbol points in the
same direction
the arrow will on
the graph.
RULE
OF THUMB:
Why do you
reverse the sign
when you multiply
or divide by a
negative number?
Here’s an example:
–1 < 5, a true
inequality statement.
If you multiply both
sides by –1, you
get 1 < –5, which
is false—a positive
number can’t be less
than a negative
number. Reversing
the inequality
symbol xes the
problem.
Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
131
7.11 Solve the inequality 2x + 3 ≥ 8x – (x – 1) for x.
Expand the right side of the inequality and simplify it.
Subtract the terms 7x and 3 from both sides of the inequality to separate the x-
terms and constants on opposite sides of the inequality symbol.
Divide both sides of the inequality by –5 to solve for x. Recall that dividing by a
negative number requires you to reverse the inequality symbol.
7.12 Solve the inequality for x.
Take the reciprocal of both sides of the inequality and reverse the inequality
sign (as directed by Problem 7.8).
Write the left side of the inequality as the product of x and a coefficient.
Multiply both sides of the inequality by 9, the reciprocal of the x-coefficient.
Divide the numerator and denominator by 3, the greatest common factor of 12
and 63, to reduce the fraction to lowest terms.
You can’t solve
for x until it’s in the
numerator, whether it’s
an inequality like this
one or an equation like
in Problem 4.14.
Why reverse the
inequality sign?
Here’s an example:
, but if you take
the reciprocal of both
sides, you get 4 < 2,
which is false. You have
to reverse the sign to
x the problem: 4 > 2.
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