Solve the Inequality for z -5z+1< -14

Solution:

Step 1: Isolate the variable term

Move all terms not containing $z$ to the opposite side of the inequality.

Step 1.1: Isolate the inequality

Subtract $1$ from both sides to get $-5z<-15$.

Step 1.2: Simplify the inequality

Combine like terms to simplify the inequality to $-5z<-15$.

Step 2: Solve for $z$

Divide the inequality by the coefficient of $z$ to find its value.

Step 2.1: Invert the inequality

When dividing by a negative number, reverse the inequality sign to get $z>3$.

Step 2.2: Simplify the left-hand side

Eliminate the common factor to isolate $z$.

Step 2.2.1: Factor cancellation

Cancel out $-5$ from both sides to leave $z$ on the left.

Step 2.2.1.1: Cancel the common factor

Perform the cancellation to get $z>\frac{-15}{-5}$.

Step 2.2.1.2: Simplify the variable term

Divide $z$ by $1$ to maintain the inequality $z>3$.

Step 2.3: Simplify the right-hand side

Perform the division to find the value of $z$.

Step 2.3.1: Execute the division

Divide $-15$ by $-5$ to get $z>3$.

Step 3: Express the solution

Write the solution in different notations.

Inequality Form: $z>3$ Interval Notation: $(3,\mathrm{\infty })$

Knowledge Notes:

The problem involves solving an inequality, which is similar to solving equations but with special attention to the direction of the inequality sign. Here are some relevant knowledge points:

1. Moving Terms: When solving inequalities, it's common to move terms to one side to isolate the variable. This is done by performing the same operation on both sides of the inequality.

2. Inequality Direction: The direction of an inequality sign must be reversed when both sides are multiplied or divided by a negative number. This is because multiplying or dividing by a negative number reverses the order of the numbers.

3. Simplifying Expressions: Simplifying the inequality involves combining like terms and reducing fractions if possible.

4. Division by Negative Numbers: When dividing by a negative number, the inequality sign flips. For example, if you have $-a<b$ and you divide both sides by $-1$, you get $a>-b$.

5. Interval Notation: This is a way of writing the set of solutions to an inequality. For an inequality like $z>3$, the interval notation is $(3,\mathrm{\infty })$, which means that $z$ can be any number greater than 3 but not including 3 itself.

6. Checking Solutions: It's always a good practice to check if the solution makes sense by plugging it back into the original inequality.

7. Representation of Solutions: Solutions to inequalities can be represented in various forms, including inequality notation, interval notation, and graphically on a number line.