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Chapter 2: Problem 42

Solve the linear equation \(5(x+3)-2(x-4)=2(x+7),\) and graph the solution set on a number line. Solve the linear inequality \(5(x+3)-2(x-4)>2(x+7),\) and graph the solution set on a number line.

Short Answer

Expert verified
For the equation, x = -9. For the inequality, x > -9.

Step by step solution

01

- Expand the Equation

Distribute the terms within the parentheses.For the equation: - Left Side: \[5(x+3)-2(x-4)= 5x + 15 - 2x + 8 \] Simplified: \[3x + 23\] - Right Side: \[2(x+7)=2x + 14\]So the equation becomes: \[3x + 23 = 2x + 14\]
02

- Isolate the Variable

Move all the terms involving x to one side by subtracting 2x from both sides. \[3x + 23 - 2x = 2x + 14 - 2x\]This simplifies to: \[x + 23 = 14\]
03

- Solve for x

Isolate x by subtracting 23 from both sides. \[x + 23 - 23 = 14 - 23\] \[x = -9\]
04

- Graph the Solution on a Number Line

Plot the point x = -9 on a number line with a solid dot to represent that the solution is exactly at x = -9. Label this point appropriately.
05

- Solve the Inequality

Use the same strategy as for the equation but without the equality.Starting with: \[5(x+3) - 2(x-4) > 2(x+7)\]Expand the terms: \[5x + 15 - 2x + 8 > 2x + 14\]Simplify to: \[3x + 23 > 2x + 14\]
06

- Isolate the Variable in the Inequality

Move all the terms involving x to one side by subtracting 2x from both sides. \[3x + 23 - 2x > 2x + 14 - 2x\]This simplifies to: \[x + 23 > 14\]
07

- Solve the Inequality

Isolate x by subtracting 23 from both sides. \[x + 23 - 23 > 14 - 23\] \[x > -9\]
08

- Graph the Solution on a Number Line for the Inequality

Graph the solution set. Since x > -9, use an open circle at -9 and shade the number line to the right to indicate all values greater than -9.

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Key Concepts

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

linear equations
Linear equations are the foundation of algebra. They involve expressions that equate to a constant value. An example is the equation from the exercise, which is simplified step-by-step.

  • First, distribute the constants inside the parentheses.
  • Next, combine like terms on both sides.
  • Finally, solve for the variable by isolating it.

Here is the breakdown:

  • Starting with the equation: \[5(x+3) - 2(x-4)= 2(x+7)\]
  • Distribute the constants: \[5x + 15 - 2x - 8 = 2x + 14\]
  • Simplify to: \[3x + 23 = 2x + 14\]
  • Isolate the variable by subtracting 2x: \[x + 23 = 14\]
  • Solve for \[x\]: \[x = -9\]

So, the solution to the linear equation is \[x = -9\].
linear inequalities
Linear inequalities are similar to linear equations, but instead of an equal sign, they use inequality symbols such as \(<, >, \le, \ge\). To solve a linear inequality, follow similar steps as solving a linear equation. One key difference is the graphical representation on the number line and the inequality symbol.

In our exercise, we start with the inequality:

  • Start with: \[5(x+3) - 2(x-4) > 2(x+7)\]
  • Distribute the constants: \[5x + 15 - 2x - 8 > 2x + 14\]
  • Simplify to: \[3x + 23 > 2x + 14\]
  • Isolate the variable \[x\] by subtracting 2x: \[x + 23 > 14\]
  • Solving gives: \[x > -9\]

The solution \[x > -9\] means that \[x\] can be any value greater than -9.
graphing solutions
Graphing solutions helps visualize where the solutions to equations and inequalities lie on the number line.

For the linear equation \[x = -9\], the point on the number line is exactly at \[x = -9\], represented with a solid dot.

  • Draw a number line.
  • Place a solid dot at -9.
  • This shows that the solution to the equation is \([-9]\).\

For the inequality \[x > -9\]:

  • Draw a number line.
  • Place an open circle at -9 (indicating that -9 is not included in the solution).
  • Shade the number line to the right of -9 to show that all numbers greater than -9 are included.

Graphing makes it clear and easy to see the range of possible solutions.

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Most popular questions from this chapter

For each compound inequality, decide whether intersection or union should be used. Then give the solution set in both interval and graph form. See Examples \(2-4\) and \(6-8 .\) $$ x<4 \text { or } x<-2 $$

Ted Williams and Rogers Hornsby were two great hitters in Major League Baseball. Together, they got 5584 hits in their careers. Hornsby got 276 more hits than Williams. How many base hits did each get? (Source: Neft, D. S., and R. M. Cohen, The Sports Encyclopedia: Baseball, St. Martins Griffin; New York, \(2007 .)\)

Jennifer Siegel invested some money at \(4.5 \%\) simple interest and \(\$ 1000\) less than twice this amount at \(3 \% .\) Her total annual income from the interest was \(\$ 1020 .\) How much was invested at each rate?

When a consumer loan is paid off ahead of schedule, the finance charge is less than if the loan were paid off over its scheduled life. By one method, called the rule of \(78,\) the amount of unearned interest (the finance charge that need not be paid) is given by $$ u=f \cdot \frac{k(k+1)}{n(n+1)} $$ In the formula, u is the amount of unearned interest (money saved) when a loan scheduled to run for n payments is paid off k payments ahead of schedule. The total scheduled finance charge is \(f .\) Use the formula for the rule of 78 to work Exercises \(37-40\) The finance charge on a loan taken out by Kha Le is \(\$ 380.50 .\) If 24 equal monthly installments were needed to repay the loan, and the loan is paid in full with 8 months remaining, find the amount of unearned interest.

Answer the questions in Exercises \(5-8 .\) In a motion problem the rate is given as \(x\) mph and the time is given as 10 min. What variable expression represents the distance in miles?
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