Which graph shows the solution to the inequality $-3x - 7 < 20$?
Answer (2)
To solve the inequality − 3 x − 7 < 20 , we find that -9"> x > − 9 . This is represented on a number line by an open circle at -9 and shading to the right. Therefore, the solution includes all values greater than -9.
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Add 7 to both sides: − 3 x < 27 .
Divide both sides by -3 (and reverse the inequality sign): -9"> x > − 9 .
The solution is all x greater than -9.
Graphically, this is an open circle at -9 with shading to the right: -9}"> x > − 9 .
Explanation
Analyzing the Inequality Let's solve the inequality step by step to find the range of values for x that satisfy the condition − 3 x − 7 < 20 .
Adding 7 to Both Sides First, we want to isolate the term with x . To do this, we add 7 to both sides of the inequality: − 3 x − 7 + 7 < 20 + 7
Simplifying the Inequality This simplifies to: − 3 x < 27
Dividing by -3 and Reversing the Inequality Now, we need to isolate x by dividing both sides by -3. Remember that when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign: \frac{27}{-3}"> − 3 − 3 x > − 3 27
Solution for x This simplifies to: -9"> x > − 9
Graphical Representation The solution to the inequality is -9"> x > − 9 . This means that x can be any number greater than -9, but not equal to -9. On a number line, this is represented by an open circle at -9 (to indicate that -9 is not included) and shading to the right (to indicate all numbers greater than -9).
Examples
Imagine you're setting a maximum spending limit. If you have a budget constraint represented by an inequality, solving it tells you the range of possible expenses you can afford. For example, if − 3 x − 7 < 20 represents your spending limit where x is the number of items you buy, solving it ( -9"> x > − 9 ) tells you how many items you can purchase without exceeding your budget. This concept is useful in personal finance, business planning, and resource management.
