Solving Inequality Equations

Inequality Equations; what are they?

It's an equation written with a "less than" (\(<, \leq\)) or "greater than" (\(>, \geq \)) symbol where you might normally see an equals (\(=\)) sign. Here we're going to look at how to solve these for a few different cases. You will need to be comfortable with:

Solving a Linear Inequality

Let's take the equation \(2x - 6 = 4\). You might be able to do this in your head, but let's write out the steps: \[\begin{align} 2x-6&=4 \\2x&=10 \\x&=5 \end{align}\] Now, if we were to be asked to solve \(2x - 6 > 4\), you might notice some similarities! \[\begin{align} 2x-6&>4 \\2x&>10 \\x&>5 \end{align}\]

Rules and Warnings

Easy right?! Well let's run through some things to be careful of:
1) \(x=5\) and \(5=x\) are the same. \(x>5\) and \(5>x\) are not!
2) \(x=5\) and \(-x=-5\) are the same. We have to flip the symbol with inequalities when multiplying by \(-1\). Let's carefully rearrange an example to see why: \[\begin{align} x&>5 \\x - 5 &> 5-5 \\x - 5 - x &> 0 - x \\-5 &> -x \\-x &< -5 \end{align}\] This becomes especially important later on with quadratics...

Quadratic Inequality Worked Solution

Leading on from the above example, we're going to solve a harmless looking quadratic \[x^2 + 3x + 2 \leq 0\] If we were to solve this as if it was an equality, we could by factorising: \[\begin{align} x^2 + 3x + 2 &= 0 \\(x+2)(x+1) &= 0 \\x = -2 \text{ or } x &= -1 \end{align}\] So where does the \(\leq\) symbol fit in? Do we just stick it in to both solutions maybe so \(x \leq -2\) and \(x \leq -1\)? Nope! We need to do a sketch.

graph of quadratic

The inequality in this question means "Less than or equal to". So the whole equation means "The parts of the function \(y = x^2 + 3x + 2\) where \(y\) is less than or equal to 0". From our sketch, we can see that it goes below \(y = 0\) (the \(x\) axis) between our two root values! These are highlighted red. So the actual answer is: \[-2\leq x \leq -1\]

Quadratic Inequality Worked Solution 2

Question: Find values for \(x\) which satisfy \[x^2 + x + 2 \geq 4\] Answer: First we solve this as if there was an equals not an inequality. \[\begin{align} x^2 + x + 2 &= 4 \\x^2 + x - 2 &= 0 \\(x-1)(x+2) &= 0 \\x = 1 \text{ or } x &= -2 \end{align}\] Now we sketch \(x^2 + x - 2 = 0\):

graph of quadratic

We know that \(x^2 + x + 2 \geq 4\) so \(x^2 + x - 2 \geq 0\) On our sketch, we see that the equation has values greater than 0 outside of our roots, on the "arms" of the curve as highlighted in red. Our solution: \[x \leq -2 \text{ or } x \geq 1\] Note: We can't combine these into one set like we did in the previous example!

Sketching Inequality Region

You may be asked specifically to do a sketch, and then to shade in the region which satisfies the equation. This is just a small extension to what we've already done, it's just a case of marking the solution on top of our sketch, like so:

shading inequality regions