Problem 137 - College Algebra Essentials

Question

Will help you prepare for the material covered in the next section. Is \(-1\) a solution of \(3-2 x \leq 11 ?\)

Step-by-Step Solution

Answer

Yes, -1 is a solution of the inequality \(3-2x \leq 11\).

Step 1: Understand the inequality

We are given the inequality \(3-2x \leq 11\). This means the expression \(3-2x\) should produce an output which is less than or equal to 11.

Step 2: Substitute -1 into the inequality

We now substitute -1 into the inequality in place of x. Doing so gives us \(3-2(-1)\) on the left-hand side.

Step 3: Evaluate \(3 - 2(-1)\)

Evaluating \(3 - 2(-1)\) results in 3 + 2 = 5.

Step 4: Check the inequality's condition

Now, we need to check if this value, 5, satisfies the condition of being less than or equal to 11. It does, so -1 is indeed a solution for the inequality.

Key Concepts

Solution Verification

One of the key aspects of solving inequalities is verifying the solution. Once you have a potential solution, like \( x = -1 \), the next step is to check if this value satisfies the original inequality. Verification involves substituting the value back into the inequality and simplifying the expression to see if it holds true.

For instance, in the inequality \( 3-2x \leq 11 \), after finding a potential solution, you replace \( x \) with \(-1\) and evaluate. If after simplification, the inequality condition (i.e., "less than or equal to 11") is still satisfied, then the chosen value is indeed a correct solution.

Solution verification is important because it acts as a confirmation step, ensuring that all transformations and manipulations have been done correctly, and that the solution accurately reflects the conditions set by the original inequality.

Substitution

Substitution is an essential part of solving inequalities and equations. It involves replacing a variable with a specific value to simplify and solve an algebraic expression.

When given the inequality \( 3-2x \leq 11 \), substituting means you replace \( x \) with \(-1\), resulting in a new expression: \( 3-2(-1) \). This particular step is crucial because it allows you to simplify the inequality into a form that can be easily evaluated.

Substitution helps in seeing if the chosen value actually satisfies the inequality. It's a practical way to test potential solutions and understand if further steps are needed to find the correct answer. Whether you are dealing with equations or inequalities, mastering substitution will significantly enhance your algebra solving skills.

Algebraic Expressions

Understanding algebraic expressions is foundational to tackling inequalities. An algebraic expression like \( 3-2x \) consists of numbers, variables, and operations (in this case, subtraction and multiplication). Such expressions are the building blocks of more complex mathematical problems.

When you are tasked with solving an inequality like \( 3-2x \leq 11 \), you are working to find values of \( x \) that make the expression true when set against a particular condition ("less than or equal to 11").

Breaking down an expression involves plugging in different values to see how they transform the expression's output. This interaction with variables and constants helps in understanding how different components influence the inequality. A firm grasp of algebraic expressions enables you to manipulate and rearrange equations or inequalities, ultimately leading you to the solution.