Solving Linear Equations and Inequalities

Properties of Inequality

Addition Property of Inequality

The addition property of inequality can be used to undo subtraction when solving inequalities.

In a linear inequality in one variable, each term can be written as a number or a product of the variable and a number. The variable is not raised to a power, in the denominator of a fraction, or under a radical such as a square root. A linear inequality is the same as a linear equation except it has an inequality symbol (<\lt, \leq, >\gt, or \geq) instead of an equal sign.

A solution of an inequality is a value of the variable that makes the inequality true, which means the comparison is valid. The process for solving linear inequalities is similar to the process for linear equations. Properties of inequality are used to isolate the variable. The difference is there can be infinitely many solutions of an inequality.

The addition property of inequality states that the solution of an inequality does not change after adding the same number to both sides of the inequality.

Addition Property of Inequality Examples
If the same quantity is added to both sides of a true inequality, the resulting inequality is still true. If a<ba \lt b, then:
a+c<b+ca+c \lt b+c
If aba\leq b, then:
a+cb+ca+c\leq b+c
If a>ba> b, then:
a+c>b+ca+c \gt b+c
If aba\geq b, then:
a+cb+ca+c\geq b+c

Step-By-Step Example
Solving an Inequality with the Addition Property
Solve and then graph the inequality on a number line:
x3<7x-3 \lt 7
Step 1
Apply the addition property of inequality to undo the subtraction. Add 3 to both sides.
x3<7x3+3<7+3\begin{aligned}x-3& \lt 7\\x-3+3& \lt 7+3 \end{aligned}
Step 2
Simplify the inequality.
x3+3<7+3x<10\begin{aligned}x-3+3& \lt 7+3\\x& \lt 10 \end{aligned}
Solution

The solutions of the inequality are all numbers less than 10. This can be expressed in different ways.

  • Inequality: x<10x \lt 10
  • Interval notation: (,10)(-\infty,10)
  • Set-builder notation: {xx<10}\{x|x \lt 10\}
Since the solutions of the inequality are less than 10, the number line should show an open dot at the number 10 with an arrow pointing toward negative infinity.
The number line shows that all numbers less than 10 are solutions to the inequality x3<7x-3 \lt 7.

Subtraction Property of Inequality

The subtraction property of inequality can be used to undo addition when solving inequalities.
The subtraction property of inequality states that the solution of an inequality does not change after subtracting the same number from both sides of the inequality. This property is analogous to the subtraction property of equality.
Subtraction Property of Inequality Examples
If the same quantity is subtracted from both sides of a true inequality, the resulting inequality is still true. If a<ba \lt b, then:
ac<bca-c\lt b-c
If aba\leq b, then:
acbca-c\leq b-c
If a>ba \gt b, then:
ac>bca-c\gt b-c
If aba\geq b, then:
acbca-c\geq b-c

Step-By-Step Example
Solving an Inequality with the Subtraction Property
Solve and then graph the inequality on a number line:
x+5>9x+5 \gt 9
Step 1
Apply the subtraction property of inequality to undo the addition. Subtract 5 from both sides.
x+5>9x+55>95\begin{aligned}x+5& \gt 9\\x+5-5& \gt 9-5\end{aligned}
Step 2
Simplify the inequality.
x+55>95x>4\begin{aligned}x+5-5& \gt 9-5\\x& \gt 4\end{aligned}
Solution

The solutions of the inequality are all numbers greater than 4. This can be expressed in different ways.

  • Inequality: x>4x \gt 4
  • Interval notation: (4,)(4,\infty)
  • Set-builder notation: {xx>4}\{x|x \gt 4\}
Since the solutions of the inequality are greater than 4, the number line should show an open dot at the number 4 with an arrow pointing toward infinity.
The number line shows that all numbers greater than 4 are solutions of the inequality x+5>9x+5 \gt 9.

Multiplication Property of Inequality

The multiplication property of inequality can be used to undo division when solving inequalities.

The multiplication property of inequality states that the solution of an inequality does not change if:

  • both sides of the inequality are multiplied by the same positive number, or
  • both sides of the inequality are multiplied by the same negative number and the inequality sign is reversed
To understand why the inequality symbol is reversed, consider an example using numbers only.
3>01(3)>?1(0)30Reverse sign3<0\begin{aligned}3&\gt 0\\-1(3)&\stackrel{?}{\gt} -1(0)\\-3&\ngtr 0 \;\;\;\;\;\;\;\;\;\;\xrightarrow{\text{Reverse sign}}\;\; -3\lt 0\end{aligned}

Notice that the positive value is greater than zero and the negative value is less than zero. Keeping the inequality sign the same would produce a false statement because 30-3\ngtr0. This is why the inequality symbol is reversed.

Multiplication Property of Inequality Example
If both sides of a true inequality are multiplied by the same positive quantity, the resulting inequality is still true. If a<ba\lt b and c>0c\gt 0, then:
ac<bcac\lt bc
If aba\leq b and c>0c\gt 0, then:
acbcac\leq bc
If a>ba\gt b and c>0c\gt 0, then:
ac>bcac\gt bc
If aba\geq b and c>0c\gt 0, then:
acbcac\geq bc
If both sides of a true inequality are multiplied by the same negative quantity and the inequality symbol is reversed, the resulting inequality is still true. If a<ba\lt b and c<0c\lt 0, then:
ac>bcac\gt bc
If aba\leq b and c<0c\lt 0, then:
acbcac\geq bc
If a>ba\gt b and c<0c\lt 0, then:
ac<bcac\lt bc
If aba\geq b and c<0c\lt 0, then:
acbcac\leq bc

Step-By-Step Example
Solving an Inequality with the Multiplication Property
Solve and then graph the inequality on a number line:
x39\frac{x}{-3}\ge-9
Step 1
Apply the multiplication property of inequality to undo the division. Multiply both sides by –3. Since 3 is negative, switch the inequality sign from \ge to \le.
x393(x3)3(9)\begin{aligned}\frac{x}{-3} &\geq -9\\-3\left(\frac{x}{-3} \right)&\leq -3\left(-9\right)\end{aligned}
Step 2
Simplify the inequality.
3(x3)3(9)x27\begin{aligned}-3\left(\frac{x}{-3} \right)&\leq -3\left(-9\right)\\x &\leq 27\end{aligned}
Solution

The solutions of the inequality are all numbers less than or equal to 27. This can be expressed in different ways:

  • Inequality: x27x \leq 27
  • Interval notation: (,27](-\infty,27]
  • Set-builder notation: {xx27}\{x|x\leq27\}
Since the solutions of the inequality are less than or equal to 27, the number line should show a closed dot at the number 27 with an arrow pointing toward negative infinity.
The number line shows that all numbers less than or equal to 27 are solutions of the inequality x39\frac{x}{-3} \geq -9.

Division Property of Inequality

The division property of inequality can be used to undo multiplication when solving inequalities.

Like the multiplication property of inequality, the division property of inequality has two parts.

The division property of inequality states that the solution of an inequality does not change if:

  • both sides of the inequality are divided by the same positive number, or
  • both sides of the inequality are multiplied by the same negative number and the inequality sign is reversed
Division Property of Inequality Example
If both sides of a true inequality are divided by the same positive quantity, the resulting inequality is still true. If a<ba\lt b and c>0c\gt 0, then:
ac<bc\frac{a}{c}\lt \frac{b}{c}

If aba\leq b and c>0c\gt 0, then:
acbc\frac{a}{c}\leq \frac{b}{c}

If a>ba>b and c>0c\gt 0, then:
ac>bc\frac{a}{c}\gt \frac{b}{c}

If aba\geq b and c>0c\gt 0, then:
acbc\frac{a}{c}\geq \frac{b}{c}
If both sides of a true inequality are divided by the same negative quantity and the inequality symbol is reversed, the resulting inequality is still true. If a<ba\lt b and c<0c\lt 0, then:
ac>bc\frac{a}{c}\gt \frac{b}{c}

If aba\leq b and c<0c\lt 0, then:
acbc\frac{a}{c}\geq \frac{b}{c}

If a>ba>b and c<0c\lt 0, then:
ac<bc\frac{a}{c}\lt \frac{b}{c}

If aba\geq b and c<0c\lt 0, then:
acbc\frac{a}{c}\leq \frac{b}{c}

Step-By-Step Example
Solving an Inequality with the Division Property
Solve and then graph the inequality on a number line:
4x124x\ge12
Step 1
Apply the division property of inequality to undo the multiplication. Divide both sides by 4.
4x124x4124\begin{aligned}4x &\geq12\\ \frac{4x}{4} &\geq\frac{12}{4}\end{aligned}
Step 2
Simplify the inequality.
4x4124x3\begin{aligned}\frac{4x}{4} &\geq\frac{12}{4}\\ x &\geq 3\end{aligned}
Solution

The solutions of the inequality are all numbers greater than or equal to 3. This can be expressed in different ways:

  • Inequality: x3x \geq 3
  • Interval notation: [3,)[3,\infty)
  • Set-builder notation: {xx3}\{x|x\geq3\}
Since the solutions of the inequality are greater than or equal to 3, the number line should show a closed dot at the number 3 with an arrow pointing toward infinity.
The number line shows that all numbers greater than or equal to 3 are solutions of the inequality 4x124x\geq12.