Lesson 66: Solving Inequalities by Adding or Subtracting

New Concept

When the same number is added to both sides of an inequality, the statement remains true. If $a > b$, then $a + c > b + c$.

What’s next

Next, you’ll apply this rule to solve for a variable in an inequality and graph the infinite solutions on a number line.

Property

For any real numbers $a$, $b$, and $c$:

• If $a < b$, then $a + c < b + c$.
• If $a > b$, then $a + c > b + c$.
• If $a \leq b$, then $a + c \leq b + c$.
• If $a \geq b$, then $a + c \geq b + c$.

Explanation

Think of an inequality as a wobbly seesaw. As long as you add the same weight (or number) to both sides, it stays tilted in the same direction! This property is your secret weapon for solving inequalities. It lets you isolate the variable by adding the same value to both sides without messing up the inequality’s truth.

Examples

• To solve $x - 12 < -8$, we add 12 to both sides: $x - 12 + 12 < -8 + 12$, which simplifies to $x < 4$.
• To solve $y - 5 \geq 2$, we use the same magic: $y - 5 + 5 \geq 2 + 5$, which simplifies to $y \geq 7$.
• For a fractional challenge like $z - \frac{1}{2} > 5$, just add $\frac{1}{2}$ to each side: $z - \frac{1}{2} + \frac{1}{2} > 5 + \frac{1}{2}$, resulting in $z > 5\frac{1}{2}$.

Just like with equations, our first step is to isolate the variable. This first key idea from our lesson, the Addition Property of Inequality, is the perfect tool for the job.

Example Problem

Solve the inequality $x - 12 < -5$ and graph the solution on a number line.

Step-by-Step

1. Start with the given inequality.

2. Use the Addition Property of Inequality by adding $12$ to both sides.

3. Simplify the expression to find the solution.

4. To graph this, we place an open circle at $7$ because $7$ is not included in the solution. We then shade the number line to the left of $7$.

Property

For any real numbers $a$, $b$, and $c$:

• If $a < b$, then $a - c < b - c$.
• If $a > b$, then $a - c > b - c$.
• If $a \leq b$, then $a - c \leq b - c$.
• If $a \geq b$, then $a - c \geq b - c$.

Explanation

Just like adding, subtracting the same amount from both sides of our trusty inequality seesaw keeps it balanced in the same way. This lets you clear out extra numbers that are added to your variable, getting you one step closer to solving the puzzle. It’s the perfect partner to the addition property, helping you isolate that sneaky variable!

Examples

• To solve $x + 7 > 10$, you subtract 7 from both sides: $x + 7 - 7 > 10 - 7$, which gives you the solution $x > 3$.
• For an inequality with decimals like $k + 3.3 \leq 5.5$, you do the same: $k + 3.3 - 3.3 \leq 5.5 - 3.3$, which simplifies to $k \leq 2.2$.
• If your allowance plus 5 dollars is at least 15 dollars ($a+5 \geq 15$), solve by subtracting 5: $a \geq 10$ dollars.

Now, let's tackle an inequality where the variable has something added to it. Here we apply the second key idea from our lesson, the Subtraction Property of Inequality.

Example Problem

Solve the inequality $y + 5 > 8$. Then graph and check the solution.

Step-by-Step

1. Begin with the inequality.

2. Apply the Subtraction Property of Inequality. Subtract $5$ from both sides.

3. Simplify to find the solution.

4. The solution is all numbers greater than $3$. We graph this with an open circle at $3$ and an arrow pointing to the right.