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Chapter 2: Problem 38

Solve each inequality and graph the solution on the number line. $$13-17 x < 34-10 x$$

Short Answer

Expert verified
The solution to the inequality is \(x > -3\). On a number line, there would be an open circle at -3 with the line shaded to the right.

Step by step solution

01

Simplify the inequality

Start by isolating the variable terms to one side of the inequality and the constant terms to the other side. Do this by first adding \(17x\) to both sides of the inequality: \(13-17x+17x < 34-10x+17x\), which simplifies to \(13 < 34+7x\).
02

Continue to isolate the variable

Move the numeric constants to one side of the inequality, separate from the variable. This is done by subtracting 34 from both sides: \(13-34 < 34-34+7x\), which simplifies to \(-21 < 7x\).
03

Solve the inequality for x

To solve for the variable, divide by the coefficient of x on both sides of the inequality: \(-21/7 < 7x/7\), which simplifies to \(-3 < x\) or in a more usual way, \(x > -3\).
04

Graph the solution on the number line

On the number line, place an open circle at -3 (since the inequality is 'greater than' and not 'greater than or equal to'). You should shade to the right of -3, indicating that any number greater than -3 is a solution to the inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Number Line Graphing
Graphing inequalities on a number line is a useful way to visually represent the solution set of an inequality. In number line graphing, you identify which values are part of the solution and which are not.

To do this, follow these steps:
  • First, locate the point on the number line which corresponds to the critical value鈥攊n this case, -3.
  • Since the inequality here is 'greater than', we use an open circle to represent that -3 is not part of the solution.
  • Next, shade or draw an arrow to the right of -3 on the number line, to indicate that all numbers greater than -3 satisfy the inequality.
Number line graphing makes it clear which numbers satisfy the inequality and is a great way to check your solution visually.
Variable Isolation
Variable isolation is a key technique used in solving inequalities and equations. It involves manipulating the inequality to have the variable you are solving for on one side alone.

To isolate the variable in this inequality, follow these steps:
  • Move all terms containing the variable to one side. For example, by adding 17x to both sides, any negative variable terms on one side can be eliminated.
  • Then, remove any constants from the side of the inequality that contains the variable by adding or subtracting them from both sides.
The goal is to have just the variable term left on one side of the equation. This makes it easier to see the solution once the inequality simplifies.
Algebraic Simplification
Algebraic simplification is the process of rewriting an expression in a simpler form. This often involves combining like terms and reducing equations or inequalities to their simplest form.

In the context of solving inequalities, simplification enables clearer understanding and handling of the expressions involved:
  • First, perform any operations that simplify the inequality. Here, combining like terms by adding 17x to both sides meant that terms with the variable were consolidated.
  • Subtract or add constants to simplify the expression further, like subtracting 34 from both sides.
  • Finally, solve for the variable by performing division or multiplication, which was done in this case by dividing both sides by 7.
  • After simplification, if you have the form of the inequality like \(x > -3\), you know what values of x satisfy the inequality.
Simplification can avoid mistakes and often highlights the solution path, making it an indispensable part of solving algebraic equations and inequalities.

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Most popular questions from this chapter

Solve each inequality and graph the solution on the number line. $$-4 \leq-4 x < 12$$

Create a mixture problem that could be solved by using the equation \(4 x+6(12-x)=5(12)\).

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Create a mixture problem of your own, and solve it.

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