Solve this inequality 8z+3-2z 51 – In the realm of mathematics, inequalities present intriguing challenges that require a systematic approach to unravel their solutions. Embark on a journey to conquer the inequality 8z+3-2z ≤ 51, a mathematical puzzle that will test your algebraic prowess and deepen your understanding of this fundamental concept.
Delving into the intricacies of inequalities, we will uncover their properties, explore diverse types, and meticulously dissect the steps involved in solving them. Through a blend of clear explanations and engaging examples, this discourse will illuminate the path to mastering inequalities.
Inequality Basics
Inequalities are mathematical statements that compare two expressions using symbols like greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). They describe relationships between quantities that are not necessarily equal.
Different types of inequalities include:
- Strict inequalities: >,<
- Non-strict inequalities: ≥, ≤
- Compound inequalities: Combinations of inequalities using “and” or “or”
To solve an inequality, isolate the variable on one side of the inequality symbol by performing algebraic operations such as adding, subtracting, multiplying, or dividing both sides by the same number.
Solving the Given Inequality
Identifying the Inequality
The given inequality is 8z+3-2z ≤ 51.
Simplifying the Inequality
Combining like terms, we get 6z+3 ≤ 51.
Isolating the Variable, Solve this inequality 8z+3-2z 51
Subtracting 3 from both sides, we get 6z ≤ 48.
Dividing both sides by 6, we get z ≤ 8.
Solving for z
Therefore, the solution to the inequality is z ≤ 8.
Checking the Solution: Solve This Inequality 8z+3-2z 51
Substituting z = 8 back into the original inequality, we get 8(8)+3-2(8) ≤ 51.
Simplifying, we get 64+3-16 ≤ 51, which is true.
Therefore, the solution z ≤ 8 is correct.
Checking solutions for inequalities is important to ensure that the solution satisfies the original inequality.
Graphical Representation
On a number line, we plot a closed circle at z = 8 and shade the region to the left of it.
This graphical representation shows that all values of z less than or equal to 8 satisfy the inequality.
Top FAQs
What is the first step in solving an inequality?
Identify the given inequality and simplify it by combining like terms.
How do you isolate the variable in an inequality?
Perform algebraic operations on both sides of the inequality to isolate the variable on one side.
What is the purpose of checking the solution to an inequality?
To verify if the obtained value of the variable satisfies the original inequality.
