Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$5+5 x \geq 10 ; 1$$

Short answer

Yes, 1 is a solution to the given inequality.

Step-by-step solution

Substitute the given test number into the inequality

Replace the variable \(x\) with the given number \(1\) in the inequality \(5+5x \geq 10\). This would give us \(5+5(1) \geq 10\) or \(5+5 \geq 10\)

Simplify the inequality

Resolving the calculations on the left hand side, the inequality becomes \(10 \geq 10\)

Check the validity of the inequality

As per inequality rules, the term \(10\) is indeed greater than or equal to \(10\). Hence, the provided value is a valid solution to the given inequality.

Key concepts

Inequality Solving

When it comes to inequality solving, the concept is quite similar to solving equations, but with a key difference: instead of finding a specific number, we're looking for a range of numbers that make the inequality true. An inequality tells us how one expression is related to another with respect to size, using symbols like greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤).

To verify a solution to an inequality, we substitute the proposed solution into the inequality and check if the resulting statement is true. For instance, given the inequality \(5+5x \geq 10\), to check if \(x=1\) is a solution, we substitute 1 for \(x\) and see if the inequality holds true. If, after the substitution, the left side of the inequality is greater than or equal to the right side, as it is with \(10 \geq 10\), then the number satisfies the inequality and is considered a solution.

It's crucial to remember that in inequalities involving multiplication or division by a negative number, we must flip the direction of the inequality sign—a step that's not necessary when solving equalities.

Substitution Method

The substitution method is a foundational tool in algebra, particularly when working with inequalities or equations. It involves replacing variables with numbers or other expressions to simplify a problem or to check solutions. In our example, to check if \(1\) is a solution for \(5+5x \geq 10\), we perform substitution by replacing \(x\) with \(1\).

After substitution, the process of solving continues as it would with a regular equation. We combine like terms and perform any necessary arithmetic operations. Here, when \(x\) is replaced by \(1\), we get \(5 + 5 \times 1 \geq 10\), which simplifies to \(10 \geq 10\). This confirms that our initial substitution results in a true statement and justifies \(1\) as a valid solution to the inequality.

Common Mistakes in Substitution

Students often forget to apply operations to both sides of an inequality or may make errors during simplification. It's important to follow each step carefully and double-check work to ensure accuracy in the substitution method.

Algebraic Expressions

At the heart of algebra lie algebraic expressions, which are combinations of numbers, variables, and operation symbols representing a value. These expressions form the building blocks for equations and inequalities. In the context of our inequality \(5+5x \geq 10\), there are two expressions: one on each side of the inequality sign.

The left side, \(5+5x\), combines the constant \(5\), the coefficient \(5\) (which multiplies the variable \(x\)), and the variable \(x\) itself into a single algebraic expression. The right side is a simpler expression, consisting only of the constant \(10\). When we perform operations like the substitution method, we manipulate these expressions to discover relationships between variables and constants.

Understanding and Working with Expressions

It's essential for students to understand how to manipulate algebraic expressions correctly. This includes combining like terms, distributing multiplication over addition or subtraction, and being able to recognize equivalent expressions, all of which are skills that empower us to solve a wide array of algebraic problems.