Problem 6 - Algebra 1: Concepts and Skills

Question

Determine whether the inequality is a multi-step inequality. Then explain how you would solve the inequality. $$ 4 y-3<13 $$

Step-by-Step Solution

Answer

The inequality \( 4y - 3 < 13 \) is a multi-step inequality. The solution is \( y < 4 \).

Step 1: Identify the Inequality Type

Observe the inequality \( 4y - 3 < 13 \). It requires more than one operation to isolate the variable, hence it is a multi-step inequality.

Step 2: Isolate the Variable - Addition/Subtraction

Add 3 to both sides of the inequality to cancel out the '-3' on the left side. This leads to the new inequality \( 4y < 13 + 3 \) or \( 4y < 16 \).

Step 3: Isolate the Variable - Multiplication/Division

Divide both sides of the inequality by 4 (which is the coefficient of \( y \)) to completely isolate \( y \). This results in \( y < \frac{16}{4} \).

Key Concepts

Solving Inequalities

Solving inequalities involves finding the value of a variable that satisfies an inequality statement, like "less than" or "greater than." It's much like solving for variables in equations, but with a twist. For inequalities, the solution isn't just one number—it's a set of numbers.

When tackling multi-step inequalities, you will perform several operations. Always keep the inequality balanced by performing the same operation on both sides, just as you would with equations. However, be aware of special rules that apply to inequalities, such as the effect of multiplying or dividing by negative numbers. These rules can change the direction of the inequality sign.

Whether it’s adding, subtracting, multiplying, or dividing, each step aims to gradually isolate the variable on one side of the inequality sign until the solution becomes clear.

Isolate Variable

The core objective when solving inequalities is to get the variable by itself on one side of the inequality sign. This process is known as isolating the variable. Start by eliminating any numbers or expressions that are added or subtracted from the variable, and then remove any coefficients that multiply or divide the variable.

Here's a simple rundown of how you might typically isolate a variable:

• Identify parts of the inequality that aren't part of the variable itself, like constants or coefficients.
• Use addition or subtraction to neutralize these numbers, effectively moving them to the other side of the inequality.
• Finally, deal with any coefficients attached to the variable by using multiplication or division.

By following these steps thoughtfully, you can reveal the solution where the variable stands alone and states directly which values satisfy the inequality.

Addition and Subtraction in Inequalities

In solving inequalities, one of the first steps often involves addition or subtraction. These operations help in moving constants across the inequality sign to simplify the expression. For instance, if you have an inequality like \( 4y - 3 < 13 \), your aim is to get rid of the -3.

To do this, add 3 to both sides, resulting in \( 4y < 16 \). This process doesn't change the inequality sign because you are applying the same operation across both sides. The simplicity of these operations makes them a foundational step in working through inequalities.

Just remember, always maintain the balance; whatever you add or subtract on one side must be mirrored on the other.

Multiplication and Division in Inequalities

After using addition or subtraction, the next logical step in solving many inequalities is multiplication or division. These are used to eliminate any coefficients in front of the variable, effectively isolating it. In our previous example of \( 4y < 16 \), dividing both sides by 4 simplifies it to \( y < 4 \).

However, a special rule to remember is: if you multiply or divide by a negative number, the inequality sign flips! For example, if you had to divide by -4 instead, fear not—flip the < sign to >.

This subtlety is crucial because it ensures the inequality remains true. Multistep inequalities such as this require careful attention to detail in every operation, ensuring that the set of solutions is accurately maintained.