Solve the Inequality for z -5z+1< -14
The given problem is asking for the solution to an algebraic inequality. Specifically, you are asked to find the range of values that the variable $z$can take such that the expression $-5z + 1$is less than $-14$. In other words, you need to manipulate the inequality to isolate $z$on one side and find what values it can be to satisfy the inequality condition presented. This will likely involve basic algebraic steps such as adding or subtracting the same value from both sides of the inequality and then dividing through by the coefficient of $z$to solve for the variable.
$- 5 z + 1 < - 14$
Answer
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: $(3, \infty)$
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:
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.
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.
Simplifying Expressions: Simplifying the inequality involves combining like terms and reducing fractions if possible.
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 $a > -b$.
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 $(3, \infty)$, which means that $z$ can be any number greater than 3 but not including 3 itself.
Checking Solutions: It's always a good practice to check if the solution makes sense by plugging it back into the original inequality.
Representation of Solutions: Solutions to inequalities can be represented in various forms, including inequality notation, interval notation, and graphically on a number line.
