Solving inequalities with equivalent steps

When rearranging inequalities we use equivalent steps, which do not change whether the inequality is true. The point of equivalent steps is to bring the inequality into a simpler form from which we can work out the solution.

What is an inequality

Let's take two functions f(x) and g(x). An inequality is an expression in one of these forms:

$$\begin{array}{rcl}f(x)<g(x),& & f(x)>g(x)\\ f(x)\le g(x),& & f(x)\ge g(x)\end{array}$$

The solutions of an inequality are all the values of x in (usually) the set of real numbers for which the inequality is true. An example of an inequality is 6x<12. Here f(x) = 6x and g(x) = 12. 6x is the left-hand side of the inequality and 12 is the right-hand side. The solutions of this inequality are all the values of x less than two: x<2. That is, the interval (−∞, 2).

Swapping sides and reversing the sign

With equations we could swap the right-hand side and the left-hand side without worry. We cannot do this with inequalities. For example, if we have the inequality x<5, then 5 < x cannot be true at the same time. In the first case x is less than five, in the second x is greater than five. However, if we reverse the inequality sign at the same time as swapping the sides, the step is equivalent.

So if we have the inequality f(x) < g(x), we can also write g(x) > f(x). We swapped the sides, but at the same time we wrote > instead of <. Similarly for less than or equal to: we can rewrite the inequality f(x) ≤ g(x) as g(x) ≥ f(x).

Adding an expression

We can add to an inequality a number, or a function that is defined on the same set on which we are solving the inequality. For example, take the inequality x>0. We can add the number three to it. We get the inequality x + 3>3. That makes sense.

Similarly, we can add a whole function. If we have the inequality x > −3x + 7, we can add 3x to it and we get:

$$\begin{array}{rcl}x& >& -3x+7(+3x)\\ x+3x& >& -3x+3x+7\\ 4x& >& 7\end{array}$$

Of course we can also subtract an expression – in other words, we can add a negative number.

Multiplying an inequality by a positive expression

We can no longer just multiply an inequality by any expression, as we could with equations. If we multiply by an expression that is always positive, that is fine. But if we multiply by a negative expression, we must reverse the inequality sign (see the next section). And of course, just as with equations, we must not multiply by an expression that is zero!

What does “always positive” mean? The simplest expression that is always positive is a positive number. So if we have the inequality x>2, we can multiply it by ten, which is a positive number, and the inequality stays true. We get the inequality 10x>20. Of course we can also divide by a positive number, so from this new inequality we can divide by five to get the inequality 2x>4.

But we cannot simply multiply an inequality by the variable x, because in general the variable x can take both positive and negative values, and that is not allowed. There are, however, situations where x will always be positive. For example, in a geometry problem the variable x may be the length of a line segment. A length cannot be negative (at worst it is zero), so if we know that x is equal to the length of a line segment, we can multiply the inequality by x.

Another expression that is always non-negative is a square. If you square a number, the result is positive (or zero). So if we have the inequality

$$\frac{1}{x^2}>2,$$

we can multiply it by x². We get: 1>2x². This is, of course, provided that x is not zero. There are more functions that are always positive. The trigonometric functions sine and cosine have the range [−1, 1]. That is not always positive. But if we add two to them, we get the range [1, 3], which is an interval of positive numbers. We could multiply an inequality by such a function. So if we had the inequality

$$\frac{x+3}{\cos (x)+3}\le 3x-5,$$

we can multiply the whole thing by cos(x) + 3, because that is always a positive function. We get:

$$x+3\le (3x-5)(\cos (x)+3).$$

Multiplying an inequality by an always negative expression

With negative numbers and expressions the situation is more complicated. Look at this example: 1<2. This inequality is certainly true. What if we multiply the whole thing by minus one? We get −1<−2. Is this inequality true? No, minus two is less than minus one. So multiplying by a negative number is clearly not an equivalent step for an inequality.

However, if we reverse the inequality sign at the same time as multiplying by a negative number, the step will be equivalent. If we multiply the inequality 1<2 by −1 and at the same time write > instead of <, we have made an equivalent step and we get the inequality −1>−2.

Multiplying by negative numbers is easy; multiplying by expressions (functions) is trickier. Why can't we multiply an inequality by the variable x? Because if the variable took positive values, the sign would not change. If it took negative values, we would have to reverse the sign. In general both cases can occur: if we don't restrict x in any way, the variable can be both negative and positive. For example:

$$\frac{1}{x}>1$$

A wrong approach would be to multiply the inequality by the variable x. We would get 1>x, so the solution would be all the x that are less than one (of course still with the condition that x is not zero). But what if we substitute −1 for x in the original inequality? We get:

$$\begin{array}{rcl}\frac{1}{-1}& >& 1\\ -1& >& 1\end{array}$$

which is certainly not true. That is why we cannot just multiply an inequality by the variable x.

However, just as in the previous section we had expressions that are always positive, we can have expressions that are always negative. The easiest way to get them is to put a minus sign in front of an always positive expression. In the previous section we saw that the function cos(x) + 3 is always positive. If we put a minus sign in front of the whole expression, we get an always negative function: −(cos(x) + 3).

Raising an inequality to a positive integer power and taking roots

Raising an inequality to a power is not in general an equivalent step, just as with equations. Look at the following inequality: x>−2. What if we square it? We get: x²>4. Is that an equivalent inequality? What happens if we substitute zero for x in both inequalities? In the first case we have 0>−2, which is true. In the second case we have 0>4, which is not true. So this step was not equivalent.

When can we raise an inequality to a positive integer power? When both sides are non-negative (positive or zero). Then there is no problem. For example:

$$\begin{array}{rcl}5& >& 2{(}^2)\\ 25& >& 4\end{array}$$

Another example is the absolute value, which gives us a non-negative number. So we can safely square this inequality:

$$\begin{array}{rcl}|x|& <& 3{(}^2)\\ |x{|}^2& <& 9\end{array}$$

A root behaves in exactly the same way as a power. If both sides are non-negative, we can take a root of the inequality, where the index of the root is a positive integer. So we can nicely take the root of the previous inequality to get back to the original form:

$$\begin{array}{rcl}|x{|}^2& <& 9(\sqrt{})\\ |x|& <& 3\end{array}$$

Removing unnecessary expressions

We often solve an inequality of the form f(x)>0. In that case the function f may be complicated and could be simplified by dropping some expressions. Look at the following inequality: 2x>0. This inequality clearly has the solution x>0. If x is negative, then 2x is negative too. If you multiply a negative number by two, the number stays negative. The 2 isn't strong enough to change the sign of the expression that follows it.

We can also try to solve the inequalities 5x>0, 157x>0 and π x>0. In all these cases the solution will be the same, x>0. A positive coefficient in front of x does not change whether the inequality is true.

Formally, we get rid of these coefficients by dividing the inequality by the coefficient, like this:

$$\begin{array}{rcl}2x& >& 0(÷2)\\ x& >& 0\end{array}$$

Note that a positive coefficient (let's call it a) can only be removed if it multiplies the whole left-hand side. We cannot remove the 2 from this inequality: 2x + 7>0, because the 2 does not multiply the whole left-hand side. But we can divide the whole inequality by two and we get the inequality x + 3.5>0.

Similarly, take this example:

$$x^2\cdot (x-3)>0$$

It may look complicated, but the expression x² is always non-negative, so it won't change the sign of the bracket that follows. So we can remove this expression from the inequality – we divide the inequality by x² and we get:

$$x-3>0$$

We only have to add the condition that x² ≠ 0, because we cannot divide by zero. However, x² is zero only when x is zero, and for x = 0 the inequality is not satisfied. We get (x = 0): 0(0 − 3)>0 and 0>0 is not true.

This is not an extra method, just a handy application of the previous point about multiplying and dividing by a positive expression.