Solve Inequalities with Addition or Subtraction — Examples

Solving Inequalities by Adding

Inequalities are very much like equations. If you want to add or subtract from one side of the equation, you must perform the same operation to the other side of the equation.

When solving inequalities by adding, our goal is to have the variable on its own. This is just the same as solving an equation.

Adding a number from both sides of the inequality does not affect the direction of the inequality. However, switching left and right sides of the equation will change the direction of the inequality. For example, $x<3$ will become $3>x$. They are equivalent, just written differently. When solving problems, you can use $<$ or $>$. It's truly just personal preference.

Example 1

Solve the following inequality. $$x - 16 > 7$$ Observe that the left side is $x-16$ and we are trying to get the variable on its own. What can we do to the $-16$ to make this happen?

We can add $16$ to both sides of the equation. Remember, $16$ and $-16$ are additive inverses. This addition moves the $16$ to the other side of the inequality. $$\begin{align} x - 16 &> 7 \\ x - 16 \,\color{#29A603}{+\,16} &> 7 \,\color{#29A603}{+\,16} \\ x &> 7+16 \\ x &> 23 \\ \end{align}$$ We are now going to take a look at some of the types of problems you might encounter.

$$\begin{align} x-5 &\ge 12 \\ x - 5 \,\color{#29A603}{+\,5} &\ge 12 \,\color{#29A603}{+\,5}\\ x &\ge 12+5 \\ x &\ge 17 \\ \end{align}$$

$$\begin{align} x-13 &< 15-2 \\ x-13 &< 13\\ x - 13 \,\color{#29A603}{+\,13} &< 13 \,\color{#29A603}{+\,13}\\ x &< 13+13 \\ x &< 26\\ \end{align}$$

Solving Inequalities by Subtracting

Inequalities act very similarly to equations. If you want to add or subtract from one side of the equation, you must perform the same operation to the other side of the equation. This ensures that the inequality is still true.

When solving inequalities by subtracting, our goal is to have the variable on its own. It can appear either on the lesser or greater side of the inequality.

Subtracting a number from both sides of the inequality does not affect the direction of the inequality. However, switching left and right sides of the equation will change the direction of the inequality.

For example, $x<3$ will become $3>x$. They are equivalent, just written differently. When solving problems, you can have the inequality pointing in either direction. It really doesn't matter.

Example 1

Solve the following inequality. $$x+3 > 10$$ Observe that the left side is $x+3$ and we are trying to get the variable on its own. That means we need to move the $+3$ somewhere else. But how?

We can subtract $3$ from both sides of the equation. $$\begin{align} x+3 &> 10 \\ x +3\,\color{#29A603}{-\,3} &> 10 \,\color{#29A603}{-\,3} \\ x&> 10-3 \\ x &> 7\\ \end{align}$$ We are now going to take a look at some of the types of problems you might encounter.

$$\begin{align} x+10 &\ge -4 \\ x +10 \,\color{#29A603}{-\,10} &\ge -4 \,\color{#29A603}{-\,10}\\ x &\ge -4-10 \\ x &\ge -14\\ \end{align}$$

$$\begin{align} 5x +3 &<4x+2\\ 5x +3 \,\color{#29A603}{-\,4x} &< 4x+2 \,\color{#29A603}{-\,4x}\\ 5x-4x+3 &<2 \\ x+3 &< 2\\ x +3 \,\color{#29A603}{-\,3} &< 2 \,\color{#29A603}{-\,3}\\ x&<2-3 \\ x&<-1 \\ \end{align}$$

Using Addition to Solve Inequalities

When we use addition to solve an inequality, it's the same as using addition to solve a regular equation! Just remember to add the same value to both sides, then simplify! The inequality sign will stay the same the entire time.

Additional Example

Let's do another example! In this example the inequality sign is $\geq$ instead of $\leq$. However, that doesn't change how you do any of the steps of the problem. All it changes is how to read the final answer.

To solve for $x,$ we simply add $15$ to both sides, then add $2$ to both sides! Remember, the inequality sign will stay the same.

$$\begin{align} -15+x -2 &\geq 41\\ -15\,{\color{#29A603}{ +\,15}}+x -2 &\geq 41\,\color{#29A603}{+\,15}\\ x -2 &\geq 56\\ x -2 \,\color{#29A603}{+\,2} &\geq 56\,\color{#29A603}{+\,2}\\ x&\geq 58 \end{align}$$

Subtracting

Like you may have seen before in using addition to solve inequalities, you can simplify inequalities by subtracting.

It's very similar to simplifying equations. We want to isolate the variable, and whatever we do to one side, we want to do the same to the other side.

Practice Problem and Solution

Let's try our hand at this problem using what we've learned. Solve the inequality for $x$:

$$x+15 \leq -60$$

Trying to add something to both sides will just make this harder, so this is where we use subtraction. Try subtracting $15$ from both sides. $$\begin{align} x+15&\leq-60\\ x+15\,{\color{#29A603}{-\,15}}&\le-60\,\color{#29A603}{-\,15}\\ x&\le-75 \end{align}$$

Notice that subtraction cancels out the $15$ on the side with the $x$, and once we subtract it from $-60$ we find that: $$x \le -75$$

(Video) Solving Basic Linear Inequalities

by PatrickJMT

Patrick JMT goes through some basic inequalities using subtraction.

$\large{\text{Summary}}$

When dealing with addition or subtraction, you solve inequalities the same way you would an equation. You're just putting a different symbol in the middle.

First Example

The first example he looks at is: $18>x+3$. We want to get the variable $x$ alone so we can subtract $3$ from both sides. $$\begin{align} 18&>x+3\\ 18\color{#29A603}{-3}&>x+3\color{#29A603}{-3}\\ 15&>x \end{align}$$

Second Example

The second example he looks at is:

$$m+1.4\ge5.6$$

This problem has a couple of decimals. As you might expect, we'll have to do some decimal subtraction. That doesn't complicate things much. The general idea of how to solve the problem is the same.

We do the same thing here. Subtract $1.4$ from both sides to get $x$ alone. $$\begin{align} m+1.4&\ge5.6\\ m+1.4\color{#29A603}{-1.4}&\ge5.6\color{#29A603}{-1.4}\\ m&\ge4.2 \end{align}$$