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Chapter 2: Problem 52

Solve each inequality and graph the solution set. \(19 \leq 5-4 x\)

Short Answer

Expert verified
The solution is \(x \leq -3.5\). Graph the solution by shading all values to the left of -3.5 on the number line.

Step by step solution

01

Isolate the variable

To solve the inequality for x, first we need to isolate the variable term on one side. Start by subtracting 5 from both sides to get rid of the constant term on the right.
02

Simplify the inequality

Subtract 5 from both sides: \[19 - 5 \leq\ 5 - 5 - 4x\] which simplifies to \[14 \leq -4x\]
03

Divide by the coefficient of x

To isolate x, divide both sides by -4. Remember that when you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality sign: \[\frac{14}{-4} \geq x\]
04

Simplify further

Simplify the division: \[-3.5 \geq x\]. This can also be written as \[x \leq -3.5\]
05

Graph the solution on a number line

Draw a number line and shade all the values to the left of -3.5, including -3.5 itself which should be denoted with a closed circle.

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Key Concepts

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

Solving inequalities
Solving inequalities is crucial in algebra. An inequality shows if one side is greater or less than the other side. To solve them, we mostly follow steps similar to solving equations:
  • We isolate the variable.
  • We simplify the inequality.
  • We handle coefficients on variables.

Certain things are different with inequalities. An important one is reversing the inequality sign when multiplying or dividing by a negative number. This step helps to maintain the correct relationship between both sides.
Graphing inequalities
Graphing inequalities helps visualize the solution.
Here鈥檚 a simple way to do it:
  • Draw a number line.
  • Locate the boundary value (here, -3.5).
  • Use a closed circle if the boundary value is included (as in \( \leq\ \) or \( \geq\ \)).
  • Shade the region representing the solution set. For \( x \leq -3.5\ \), shade the line left of -3.5.

This method gives a clear picture of where the solutions to the inequality lie.
Algebraic expressions
Algebraic expressions are foundations in solving inequalities.
They include constants, variables, and operations (like addition and subtraction). In this problem, we worked with \(19 \leq 5 - 4x\ \).
To solve:
  • We isolate and manipulate terms involving the variable.
  • Operations like subtraction and division simplify both sides.

Understanding how to handle these expressions helps in solving other inequalities in algebra.

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Most popular questions from this chapter

Show a complete solution to each problem. Millie's mix. Millie blends 3 pounds of Brazil nuts that sell for \(\$ 8\) per pound with 2 pounds of cashews that sell for S6 per pound. What should be the price per pound of the mixed nuts?

Show a complete solution to each problem. Rectangular lawn. If the perimeter of a rectangular lawn is \(116 \mathrm{m}\) and the width is \(6 \mathrm{m}\) less than the length, then what are the length and width?

Solve each equation. Practice combining some steps. Look for more efficient ways to solve each equation. $$\frac{3}{2} x=-\frac{9}{5}$$

Solve each equation. $$\frac{P+7}{3}-\frac{P-2}{5}=\frac{7}{3}-\frac{P}{15}$$

Discussion. In each case, write the resulting set of numbers in interval notation. Explain your answers. a) Every number in \((3,8)\) is multiplied by 4 b) Every number in \([-2,4)\) is multiplied by \(-5\) c) Three is added to every number in \((-3,6)\) d) Every number in \([3,9]\) is divided by \(-3 .\)
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