Solving Inequalities

Learning Outcomes

By the end of this section, you will be able to:

  • Solve single-step inequalities
  • Solve multi-step inequalities

Addition and Multiplication Properties of Inequality

Solving inequalities is very similar to solving equations. The difference is that you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. We solve inequalities using the addition property and the multiplication property.

Addition Property

Let, [latex]a,b,[/latex] and [latex]c[/latex] be real numbers.

If [latex]a

This statement also holds for [latex]a \leq b, a>b,[/latex] and [latex]a \geq b[/latex].

For example,

[latex]2<3[/latex]

If we add 5 to each side of the inequality, we have

[latex]2+5<3+5[/latex]

[latex]7<8[/latex]

which is a true statement. If we add -5 to each side of the inequality,

[latex]2+(-5)<3+(-5)[/latex]

[latex]-3<-2[/latex]

which is also true. We can add a positive or a negative number to each side of the inequality and the resulting statement is true. Since subtracting a number is the same as adding its negative, we can subtract a number from each side of an inequality and the resulting statement is true.

Remember when we interchange the expressions on each side of an inequality, the inequality is reversed.

[latex]-2>-3[/latex].

Example

Illustrate the addition property for inequalities by solving each of the following:

  1. [latex]x-3<5[/latex]
  2. [latex]7>2+x[/latex]
  3. [latex]x-2 \leq 4[/latex]

The following videos illustrate solving inequalities using the addition property.

Suppose in the example [latex]2<3[/latex] we multiplied by [latex]5[/latex]. Since [latex]2 \cdot 5=10[/latex] and [latex]3 \cdot 5=15[/latex], and [latex]10<15[/latex], the relationship between the products is the same. On the other hand, if we multiplied by [latex]-5[/latex], we would have [latex]2(-5)=-10[/latex] and [latex]3(-5)=-15[/latex], and [latex]-10> -15[/latex]. The inequality changes from less than [latex](<)[/latex] to greater than [latex](>)[/latex]. When we multiply both sides of an inequality by a negative, the inequality is reversed.

Multiplication Property

Let, [latex]a,b,[/latex] and [latex]c[/latex] be real numbers.

If [latex]a0[/latex] then [latex]a \ cdot c

If [latex]ab \cdot c[/latex].

This statement also holds for [latex]a \leq b, a>b,[/latex] and [latex]a \geq b[/latex].

Since division is the same as multiplying by the reciprocal, when we divide each side of an inequality by a positive number, the inequality stays the same. But if we divide each side of an inequality by a negative number, the inequality is reversed.

Example

Illustrate the multiplication property for inequalities by solving each of the following:

  1. [latex]3x \geq 15[/latex]
  2. [latex]2x-1 < 9[/latex]
  3. [latex]3-4x > 15[/latex]

Try It

 Solve Multi-Step Inequalities

As the previous examples have shown, we can perform the same operations on both sides of an inequality just as we do with equations. To isolate the variable and solve, we combine like terms and perform operations with the multiplication and addition properties.

Example

Solve the inequality: [latex]13 - 7x\ge 10x - 4[/latex].

Try It