Graph 2x< 7x-36
The question requires you to graph the inequality 2x < 7x - 36. This involves finding the set of x-values for which the inequality holds true and representing this set on a number line or a Cartesian coordinate system. You have to solve the inequality for x to find the range of x-values that satisfy the condition, and then show this range through shading or marking on the appropriate graph.
$2 x < 7 x - 36$
Answer
Solution:
Step 1: Isolate the variable $x$ on one side
Step 1.1: Subtract $7x$ from both sides to get all $x$ terms on one side.
$$ 2x - 7x < -36 $$
Step 1.2: Combine like terms on the left side.
$$ -5x < -36 $$
Step 2: Solve the inequality
Step 2.1: Divide the inequality by $-5$ to solve for $x$. Remember to reverse the inequality sign because we are dividing by a negative number.
$$ \frac{-5x}{-5} > \frac{-36}{-5} $$
Step 2.2: Simplify the inequality.
Step 2.2.1: Simplify the left side by canceling out the $-5$.
Step 2.2.1.1: Perform the cancellation.
$$ \frac{\cancel{-5}x}{\cancel{-5}} > \frac{-36}{-5} $$
Step 2.2.1.2: The left side simplifies to $x$.
$$ x > \frac{-36}{-5} $$
Step 2.3: Simplify the right side by dividing the negative numbers.
Step 2.3.1: A negative divided by a negative is positive.
$$ x > \frac{36}{5} $$
Knowledge Notes:
To solve an inequality similar to $2x < 7x - 36$, one must follow a systematic approach:
Isolating the variable: The first step is to get all the terms containing the variable on one side of the inequality. This often involves adding or subtracting terms from both sides.
Combining like terms: After moving terms, it's important to combine like terms to simplify the inequality.
Dividing or multiplying: To isolate the variable completely, you may need to divide or multiply both sides of the inequality by a number. When doing so with a negative number, the inequality sign must be flipped.
Simplifying expressions: After performing operations, it's crucial to simplify the expressions on both sides of the inequality, which may involve canceling out common factors or performing arithmetic on numbers.
Graphing the solution: The solution to an inequality is often a range of values. Graphing these values on a number line can help visualize the solution set.
Inequality signs: Remember that the inequality sign will change direction when both sides are multiplied or divided by a negative number. This is because reversing the division or multiplication by a negative number would reverse the order of the inequality.
Checking your solution: It's always a good practice to check if a number within your solution set satisfies the original inequality. This can help verify that the steps were performed correctly.
