Question

Check whether each ordered pair is a solution of the inequality. $$12 y-3 x \leq 3 ;(-2,4),(1,-1)$$

Short answer

The pair (-2,4) is not a solution to the inequality, but the ordered pair (1,-1) is a solution to the inequality.

Step-by-step solution

Set Up the Inequality

Initially, the inequality 12y - 3x ≤ 3 is given, where (x,y) can represent any ordered pair.

Substitute the First Pair

Substitute the first pair (-2,4) into the inequality. This means to replace every instance of x with -2 and every instance of y with 4. The inequality 12y - 3x ≤ 3 becomes \(12*4 - 3*(-2) ≤ 3\), which simplifies to 48 + 6 ≤ 3.

Evaluate the First Inequality

Evaluate the inequality 48 + 6 ≤ 3, which simplifies to 54 ≤ 3. Since 54 is not less than or equal to 3, the ordered pair (-2,4) is not a solution to the inequality.

Substitute the Second Pair

Now, substitute the second pair (1,-1) into the inequality. This means to replace every instance of x with 1 and every instance of y with -1. The inequality 12y - 3x ≤ 3 becomes \(12*(-1) - 3*1 ≤ 3\), which simplifies to -12 - 3 ≤ 3.

Evaluate the Second Inequality

Evaluate the inequality -12 - 3 ≤ 3, which simplifies to -15 ≤ 3. Since -15 is less than 3, the pair (1,-1) is a solution to the inequality.

Key concepts

Ordered Pair Solutions

When working with a system of equations or inequalities, an ordered pair \( (x, y) \) represents a potential solution. It's a pair of values that correspond to the variables of the equation or inequality. To check if an ordered pair is a solution to an inequality, we substitute these values into the inequality and determine if the resulting statement is true.

For instance, consider the inequality \( 12y - 3x \leq 3 \). To test the ordered pair \( (-2,4) \) we replace \( x \) with -2 and \( y \) with 4 in the inequality. After substitution, we assess whether the new equation holds true. If it does, then \( (-2,4) \) is a legitimate solution.

Linear Inequalities

Linear inequalities are similar to linear equations but instead of an equality sign, they include an inequality symbol such as \(<\), \(>\), \(\leq\) or \(\geq\). They represent a range of possible solutions rather than a single fixed solution. The process of solving a linear inequality is like solving a linear equation: we aim to isolate the variable on one side of the inequality.

A linear inequality like \( 12y - 3x \leq 3 \) defines a boundary on a graph. All points on one side of the boundary line, including the line itself if the inequality is 'less than or equal to' (\(\leq\)) or 'greater than or equal to' (\(\geq\)), are solutions to the inequality.

Inequality Substitution

Inequality substitution is a technique used to determine if an ordered pair satisfies an inequality. It involves replacing the variables in the inequality with the values from the ordered pair. If the result is a true statement, the ordered pair is a solution; if not, it is not a solution.

To illustrate, in the inequality \( 12y - 3x \leq 3 \) and with the ordered pair \( (1,-1) \) we substitute \( x \) with 1 and \( y \) with -1: \(12(-1) - 3(1) \leq 3\). Simplifying, \( -12 - 3 \leq 3 \), which is a true statement, indicating that \( (1,-1) \) is indeed a solution to the inequality.

Evaluating Inequalities

Evaluating inequalities entails performing operations to verify whether the inequality holds true for certain values. It's like checking your work to ensure it satisfies the original condition.

Consider the inequality \(12y - 3x \leq 3\) after substitution. For the ordered pair \( (-2,4) \) we obtained \( 48 + 6 \leq 3 \), which is false, indicating that \( (-2,4) \) is not a solution. On the other hand, the pair \( (1,-1) \) results in \( -15 \leq 3 \), which is true, verifying that \( (1,-1) \) is a solution. Evaluating involves simplifying the expression after substitution and concluding the truthfulness of the inequality.