Solving inequalities

Understanding Solutions to Inequalities

In this chapter, we focus on how to solve inequalities. The general idea is similar to solving one-variable equations, but there are a few important differences, especially when you multiply or divide by negative numbers.

An inequality is a statement comparing two expressions using symbols like $<$, $\le$, $>$, $\ge$. Solving an inequality means finding all values of the variable that make the inequality true.

Examples of inequalities:

• $x + 3 < 10$
• $5y - 2 \ge 13$
• $-2x > 6$

The solution to an inequality is usually a range (or ranges) of numbers, not just a single number.

Basic Rules for Solving Linear Inequalities

You use many of the same steps as when solving linear equations, with one key exception.

You are allowed to:

• Add or subtract the same number from both sides of the inequality.
• Multiply or divide both sides by the same positive number.
• Simplify expressions (distribute, combine like terms, etc.).

These actions do not change the direction of the inequality symbol.

However:

• If you multiply or divide both sides of an inequality by a negative number, you must reverse (flip) the inequality sign.

This is the main new rule you must keep in mind.

Why the Sign Flips with Negatives (Intuition Only)

If $2 < 5$, then multiplying both sides by $-1$ gives $-2$ and $-5$, but on the number line $-5$ is to the left of $-2$, so now $-2 > -5$. The relationship reversed.

So:
$$
2 < 5 \quad\Rightarrow\quad -2 > -5
$$

This is why multiplying or dividing by a negative reverses the inequality.

Solving Inequalities Step by Step

For linear inequalities in one variable (like $ax + b < c$), you generally:

1. Simplify each side (remove parentheses, combine like terms).
2. Get all variable terms on one side and constant terms on the other.
3. Isolate the variable by dividing or multiplying.
4. Remember to flip the inequality sign if you multiply or divide by a negative.

Example 1: No Sign Flip Needed

Solve: $x + 7 < 15$.

1. Subtract $7$ from both sides:
   $$
   x + 7 - 7 < 15 - 7
   $$
   $$
   x < 8
   $$

The solution is all real numbers less than $8$.

Example 2: With Sign Flip

Solve: $-3x \ge 6$.

1. Divide both sides by $-3$. Because $-3$ is negative, flip the sign:
   $$
   x \le \frac{6}{-3}
   $$
   $$
   x \le -2
   $$

The solution is all real numbers less than or equal to $-2$.

Checking Solutions

You can check your solution by trying values:

• Pick a number that should satisfy the inequality according to your answer.
• Substitute it into the original inequality.
• See if the resulting statement is true.

For $x \le -2$ from the previous example, try $x = -3$:

Original inequality: $-3x \ge 6$.

Substitute:
$$
-3(-3) \ge 6 \quad\Rightarrow\quad 9 \ge 6
$$
which is true, so it supports our solution.

Try a number that should not satisfy the inequality, e.g. $x = 0$:
$$
-3(0) \ge 6 \quad\Rightarrow\quad 0 \ge 6
$$
This is false, as expected.

Writing Solutions in Different Forms

Once you solve an inequality, you can express the solution in several common ways.

For example, if your answer is $x > 4$:

• Inequality form: $x > 4$
• Verbal description: “All real numbers greater than $4$”
• Interval idea (just conceptually here): values start just after $4$ and go to the right forever.

You will see more formal notation and graphing in the related chapter on graphing inequalities, so we keep the focus here on obtaining the solution.

Solving Inequalities with Fractions and Decimals

Solving inequalities that involve fractions or decimals uses the same rules. Be careful with signs and with multiplying or dividing by negative numbers.

Example 3: Fraction Coefficient (Positive)

Solve:
$$
\frac{1}{2}x - 3 \le 5
$$

1. Add $3$ to both sides:
   $$
   \frac{1}{2}x \le 8
   $$
2. Multiply both sides by $2$:
   $$
   x \le 16
   $$

No sign flip, because $2$ is positive.

Example 4: Fraction Coefficient (Negative)

Solve:
$$
-\frac{1}{4}x > 3
$$

1. Multiply both sides by $-4$. Since $-4$ is negative, flip the sign:
   $$
   x < 3 \cdot (-4)
   $$
   $$
   x < -12
   $$

Multi-Step Inequalities

Sometimes you need several steps to isolate the variable: distributing, combining like terms, and then solving.

Example 5: Distributive Property

Solve:
$$
3(x - 2) + 5 \le 2x + 9
$$

1. Distribute the $3$:
   $$
   3x - 6 + 5 \le 2x + 9
   $$
   $$
   3x - 1 \le 2x + 9
   $$
2. Subtract $2x$ from both sides:
   $$
   x - 1 \le 9
   $$
3. Add $1$ to both sides:
   $$
   x \le 10
   $$

No division by a negative was needed, so the sign stayed the same.

Example 6: Variable on Both Sides, Negative Coefficient

Solve:
$$
-5x + 4 > 3x - 12
$$

1. Add $5x$ to both sides:
   $$
   4 > 8x - 12
   $$
2. Add $12$ to both sides:
   $$
   16 > 8x
   $$
3. Divide both sides by $8$:
   $$
   2 > x
   $$

You can also rewrite this as:
$$
x < 2
$$
It means the same thing.

Notice we never divided by a negative number here, so the inequality symbol never flipped.

Inequalities With “$\le$” and “$\ge$”

The solving steps are identical for $\le$ and $\ge$. The only difference from $<$ and $>$ is whether equality is allowed.

Example:
$$
4x + 1 \ge 9
$$

1. Subtract $1$:
   $$
   4x \ge 8
   $$
2. Divide by $4$:
   $$
   x \ge 2
   $$

This includes $x = 2$ and all larger numbers.

When Inequalities Have No Solution or All Numbers Work

Sometimes, after simplifying, you end up with an inequality that does not involve the variable anymore. Then you must interpret what it means.

Example 7: No Solution

Solve:
$$
2(x + 1) > 2x + 10
$$

1. Distribute:
   $$
   2x + 2 > 2x + 10
   $$
2. Subtract $2x$ from both sides:
   $$
   2 > 10
   $$

The statement $2 > 10$ is never true. That means no value of $x$ can make the original inequality true. So there is no solution.

Example 8: All Real Numbers Are Solutions

Solve:
$$
3(x - 4) \le 3x - 12
$$

1. Distribute:
   $$
   3x - 12 \le 3x - 12
   $$
2. Subtract $3x$ from both sides:
   $$
   -12 \le -12
   $$

The statement $-12 \le -12$ is always true. That means every real number you plug into $x$ will satisfy the original inequality. The solution is: all real numbers.

Compound Inequalities (Briefly Solving)

Compound inequalities use words like “and” and “or”, and you will see more detail in related sections, but we can outline basic solving steps.

“And” Inequality (Combined Form)

For an inequality like:
$$
1 < 2x + 3 \le 9
$$

You can treat it as two inequalities at the same time:

• $1 < 2x + 3$
• $2x + 3 \le 9$

You solve them together, usually by working on the middle:

1. Subtract $3$ from all three parts:
   $$
   1 - 3 < 2x \le 9 - 3
   $$
   $$
   -2 < 2x \le 6
   $$
2. Divide all three parts by $2$ (positive, so no sign flip):
   $$
   -1 < x \le 3
   $$

So $x$ must be greater than $-1$ and less than or equal to $3$ at the same time.

“Or” Inequality (Separate Pieces)

For an “or” situation, like:
$$
3x - 2 \le -5 \quad\text{or}\quad 3x - 2 > 10
$$

You solve each inequality separately:

1. $3x - 2 \le -5$:
• Add $2$: $3x \le -3$
• Divide by $3$: $x \le -1$
2. $3x - 2 > 10$:
• Add $2$: $3x > 12$
• Divide by $3$: $x > 4$

The combined solution is:

• $x \le -1$ or $x > 4$

Common Mistakes to Avoid

• Forgetting to flip the sign when multiplying or dividing by a negative number.
• Losing the inequality symbol in the middle of steps (write it carefully each time).
• Mixing up “$<$” and “$>$”, or “$\le$” and “$\ge$”.
• Stopping at a step like $4x \ge 12$ and not finishing by dividing to isolate $x$.
• Not returning to the original inequality if checking your solution.

Practice Structure for Solving Inequalities

When you practice, try to:

• Write each step in a separate line.
• Show clearly what you add, subtract, multiply, or divide on each side.
• Circle or underline your final answer (for example, $x > -3$).
• Optionally test one or two values to see if they satisfy the original inequality.

The more carefully you practice the process, the more naturally these steps will come when solving inequalities of many kinds.