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Chapter 2: Problem 34

Graph each inequality. $$ x-1 \leq 0 $$

Short Answer

Expert verified
Shade the number line to the left of and including \( x = 1 \).

Step by step solution

01

Understand the Inequality

The given inequality is \( x - 1 \leq 0 \). This means we need to find values of \( x \) that make this inequality true.
02

Solve the Inequality

To solve for \( x \), add 1 to both sides of the inequality:\( x - 1 + 1 \leq 0 + 1 \).This simplifies to \( x \leq 1 \).
03

Graph the Inequality on a Number Line

Draw a number line. Locate the point \( x = 1 \) on the number line.
04

Shade the Solution Region

Shade all values to the left of \( x = 1 \) including the point \( x = 1 \). Use a closed circle at \( x = 1 \) to indicate that \( x = 1 \) is part of the solution.

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Key Concepts

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

solving inequalities
Inequalities express the relative size of two values. The inequality given here is \( x - 1 \leq 0 \). This inequality means we must find all \( x \) values that are less than or equal to 1.

To solve the inequality, isolate \( x \) on one side. Start by adding 1 to both sides:

\[ x - 1 + 1 \leq 0 + 1 \]

This simplifies to: \[ x \leq 1 \]

This tells us that any number less than or equal to 1 will make the inequality true. The solution set is all real numbers up to and including 1. In this form, it's easy to understand that 1 is the upper limit.
number line
A number line graphically represents solutions to inequalities. To graph \( x \leq 1 \), follow these steps:

  • Draw a horizontal line and mark a point for every integer. Make sure to include 1.
  • Locate the point for 1. This is where we focus our attention as it's the boundary for our inequality.


This visual representation helps see which numbers meet the inequality's criteria. Remember, if \( x \) can equal 1, use a closed circle or dot on the number line at this point. This signifies that 1 is included in the solution.
shading regions
Shading regions on a number line clearly indicate the set of solutions. To graph \( x \leq 1 \):

  • Place a closed circle at the point where \( x \leq 1 \).
  • Shade everything to the left of 1. This shading covers all numbers less than 1.


The reason for shading to the left is because the inequality encompasses all values lower than and equal to 1. By shading this region, it's easy to visually grasp which numbers satisfy the inequality. This method helps clarify and reinforce what the numerical solution represents.

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Most popular questions from this chapter

Graph the union of each pair of inequalities. $$ x+33 $$

Use your knowledge of the slopes of parallel and perpendicular lines. Show that \((-13,-9),(-11,-1),(2,-2),\) and (4,6) are the vertices of a parallelogram. (Hint: A parallelogram is a four-sided figure with opposite sides parallel.)

Graph each linear or constant function. Give the domain and range. $$ f(x)=-x $$

Determine whether each pair of lines is parallel, perpendicular, or neither. \(x+1=0\) and \(y-2=0\)

For each function, find (a) \(f(2)\) and (b) \(f(-1)\) $$ \begin{array}{c|c} x & y=f(x) \\ \hline 8 & 6 \\ \hline 5 & 3 \\ \hline 2 & 0 \\ \hline-1 & -3 \\ \hline-4 & -6 \end{array} $$
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