Lesson 77: Solving Two-Step and Multi-Step Inequalities

New Concept

Two-step and multi-step inequalities require more than one inverse operation to isolate the variable.

What’s next

Next, you’ll apply inverse operations and simplification techniques to solve for variables and graph the resulting solution sets on a number line.

Property

To solve a two-step inequality, you must use more than one inverse operation to isolate the variable. First, use addition or subtraction, then finish with multiplication or division.

Explanation

Think of it like a rescue mission for your variable! It's trapped by two bad guys: an addition/subtraction sidekick and a multiplication/division boss. To set the variable free, you must defeat them in reverse order. First, handle the addition or subtraction, then tackle the multiplication or division. It’s a two-step plan for mathematical victory!

Examples

$5x + 10 > 30 \implies 5x > 20 \implies x > 4$
$2y - 6 \le -18 \implies 2y \le -12 \implies y \le -6$

Property

Remember to change the direction of the inequality symbol when multiplying or dividing both sides by a negative number.

Explanation

When you multiply or divide both sides by a negative number, you're flipping the values to the opposite side of the number line. To keep the statement true, the inequality sign must do a flip too! Forgetting this is like walking backward without looking—you'll end up in the wrong place! So, remember: negative multiplier equals sign flipper!

Examples

$-4x > 24 \implies x < -6$
$15 - 5x \le 40 \implies -5x \le 25 \implies x \ge -5$

Remembering to flip the inequality sign is crucial when working with negative multipliers. Let's see this key idea of solving two-step inequalities in action.

Example Problem: Solve the inequality $12 - 4x > 32$ and graph the solution.

1. Start with the given inequality.

2. To begin isolating the variable term, subtract $12$ from both sides.

3. Divide both sides by $-4$. Because you are dividing by a negative number, you must reverse the direction of the inequality sign.

4. To graph the solution, place an open circle on $-5$ to show that $-5$ is not a solution. Then, shade the number line to the left of the circle to represent all numbers less than $-5$.

Property

To simplify an inequality with fractions, multiply every term on both sides by the Least Common Multiple (LCM) of the denominators.

Explanation

Fractions in an inequality can look messy! The secret weapon is the Least Common Multiple (LCM) of all the denominators. Multiply every single term by the LCM, and watch those pesky fractions magically disappear. This awesome trick leaves you with a much simpler integer problem to solve. It's the ultimate cleanup tool for your math homework!

Examples

$\frac{2}{3}y + \frac{1}{6} < \frac{5}{6} \implies 6(\frac{2}{3}y) + 6(\frac{1}{6}) < 6(\frac{5}{6}) \implies 4y + 1 < 5 \implies y < 1$
$\frac{1}{2}x - 1 > \frac{1}{4} \implies 4(\frac{1}{2}x) - 4(1) > 4(\frac{1}{4}) \implies 2x - 4 > 1 \implies x > \frac{5}{2}$

Property

Before using inverse operations to isolate the variable, simplify each side of the inequality. This may involve using distribution or combining like terms.

Explanation

Don't rush into solving! Always tidy up first by simplifying each side of the inequality. Use the distributive property to break open parentheses or combine any like terms you see. A little bit of cleanup at the beginning makes the final steps of isolating the variable much easier. It's like organizing your room before starting your homework!

Examples

$-7(2 - x) \ge -14^2 \implies -14 + 7x \ge -196 \implies 7x \ge -182 \implies x \ge -26$
$-10 + (-5) < -3d - 8 \implies -15 < -3d - 8 \implies -7 < -3d \implies \frac{7}{3} > d$

Before you can solve, you have to simplify. Let's tackle a multi-step inequality that needs a bit of unpacking first.

Example Problem: Solve the inequality $-5(3 - x) \ge -10^2$ and graph the solution.

1. Begin with the inequality.

2. Simplify both sides. Use the distributive property on the left side and evaluate the exponent on the right side.

3. Add $15$ to both sides to isolate the term with the variable.

4. Divide both sides by $5$. Since $5$ is a positive number, the inequality symbol does not change.

5. To graph the result, place a closed circle on $-17$ to show that it is a solution. Then, shade the number line to the right of the circle.