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Magazine Chapter 3: Problem 64
Solve the inequality and sketch the solution set on a number line. $$-1 \leq 2-3 x<11$$
Short Answer
Expert verified
-3 < x 鈮� 1
Step by step solution
01
Initial Inequality Setup
Consider the given inequality: MathfdescEq -1 - 2 math2-3x<11$$
02
Separate the Compound Inequality
Break down the given compound inequality into two separate inequalities: -1 鈮� 2 - 3x and 2 - 3x < 11.
03
Solve the First Inequality
Solve the first inequality -1 鈮� 2-3x : Subtract 2 from both sides: -1 - 2 Less2-3Less2 鈮� Less2-3x-2Simplify: -3 鈮� -3xDivide by -3 (remember to reverse the inequality sign): 1 鈮� x or Lessx鈮�1.
04
Solve the Second Inequality
Solve the second inequality 2 - 3x < 11:Subtract 2 from both sides: Less2 - 2Less3x< 11 - 2Simplify: -3x < 9Divide by -3 (remember to reverse the inequality sign):x > -3.
05
Combine the Solutions
Combine the solutions from the two inequalities: 鈭�3 < x 鈮� 1.
06
Sketch the Solution on a Number Line
Draw a number line and shade the region where -3 < x 鈮� 1:At x = -3, draw an open circle (since -3 is not included in the solution).At x = 1, draw a closed circle (since 1 is included).Shade the region between -3 and 1.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Compound Inequalities
When solving compound inequalities, you deal with two separate inequalities at the same time. Compound inequalities are typically written with two inequality signs, like this: $$-1 \leq 2 - 3x < 11$$. This means that both parts of the inequality must be true simultaneously. You must solve each part separately before combining the results at the end.
In our exercise, we separated the compound inequality into two simpler inequalities: -1 鈮� 2 - 3x and 2 - 3x < 11. Solving these smaller inequalities will help us find the range of values that satisfy the original compound inequality.
In our exercise, we separated the compound inequality into two simpler inequalities: -1 鈮� 2 - 3x and 2 - 3x < 11. Solving these smaller inequalities will help us find the range of values that satisfy the original compound inequality.
Number Line Representation
A number line is a visual way of representing solutions to inequalities. To sketch the solution set of an inequality on a number line, follow these steps:
In our problem, the solutions are combined as $$-3 < x \leq 1$$. This tells us that x can be any value between -3 and 1, but not including -3. To represent this on a number line:
- Identify the critical values that define the boundaries of the solution set.
- Use open or closed circles to indicate whether the boundary values are included in the solution.
- Shade the region on the number line that represents all the values satisfying the inequality.
In our problem, the solutions are combined as $$-3 < x \leq 1$$. This tells us that x can be any value between -3 and 1, but not including -3. To represent this on a number line:
- Draw an open circle at x = -3 (since it is not included).
- Draw a closed circle at x = 1 (since it is included).
- Shade the region between -3 and 1.
Reverse Inequality Sign
When solving inequalities, it is essential to remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign.
For instance, if you have $$-3 \leq -3x$$, and you divide both sides by -3 to solve for x, you must reverse the sign, resulting in: $$1 \geq x$$. Similarly, in the second inequality: $$-3x < 9$$, dividing both sides again by -3 gives: $$x > -3$$. Reversing the inequality sign is crucial as it changes the range of possible solutions, ensuring the final combined solution of $$-3 < x \leq 1$$ is accurate and inclusive of all valid x values.
For instance, if you have $$-3 \leq -3x$$, and you divide both sides by -3 to solve for x, you must reverse the sign, resulting in: $$1 \geq x$$. Similarly, in the second inequality: $$-3x < 9$$, dividing both sides again by -3 gives: $$x > -3$$. Reversing the inequality sign is crucial as it changes the range of possible solutions, ensuring the final combined solution of $$-3 < x \leq 1$$ is accurate and inclusive of all valid x values.
- Always double-check that you reverse the sign when dividing or multiplying by a negative.
- This step ensures that your final inequality set correctly defines the solution range.
