2. Q2a
Save videos to My Cheatsheet for later, for easy studying.
Video Solution
Q1
Q2
Q3
L1
L2
L3
Similar Question 1
<p>Is <code class='latex inline'>\displaystyle |y|=x </code> a function? Explain.</p>
Similar Question 2
<p>Describe the possible values for <code class='latex inline'>x</code> and <code class='latex inline'>y</code> when <code class='latex inline'>|x-y|>0</code>. What does it mean when <code class='latex inline'>|x-y|=0</code>? Can <code class='latex inline'>|x-y|<0</code>? Explain your reasoning. </p>
Similar Question 3
<p>The graph at the right models the distance between a roadside stand and a car traveling at a constant speed. The <code class='latex inline'>\displaystyle x </code>-axis represents time and the <code class='latex inline'>\displaystyle y </code>-axis represents distance. Which equation best represents the relation shown in the graph? <code class='latex inline'>\displaystyle \begin{array}{ll}\text { A } y=|60 x| & \text { C } y=|x|+60 \\ \text { B } y=|40 x| & \text { (D } y=|x|+40\end{array} </code></p><img src="/qimages/89619" />
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>Find each absolute value.</p><p><code class='latex inline'>|-0.85|</code></p>
<p>Simplify each expression. <code class='latex inline'>\displaystyle |30-28|-6 </code></p>
<p>a. Use a graphing calculator. Graph <code class='latex inline'>\displaystyle y_{1}=k|x| </code> and <code class='latex inline'>\displaystyle y_{2}=|k x| </code> for some positive value of <code class='latex inline'>\displaystyle k </code>.</p><p>b. Graph <code class='latex inline'>\displaystyle y_{1}=k|x| </code> and <code class='latex inline'>\displaystyle y_{2}=|k x| </code> for some negative value of <code class='latex inline'>\displaystyle k . </code></p><p>c. What conclusion can you make about the graphs of <code class='latex inline'>\displaystyle y_{1}=k|x| </code> and <code class='latex inline'>\displaystyle y_{2}=|k x| ? </code></p>
<p>Is the absolute value inequality or equation always, sometimes, or never true? Explain.</p><p><code class='latex inline'>\displaystyle -8 > |x| </code></p>
<p>Let </p><p><code class='latex inline'>\displaystyle x^{2}+13 x-30=(x+p)(x+q) </code></p><p>a. What do you know about the signs of <code class='latex inline'> p </code> and <code class='latex inline'> q </code> ?</p><p>b. Suppose <code class='latex inline'> |p|>|q| . </code> Which number, <code class='latex inline'> p </code> or <code class='latex inline'> q </code> , is a negative integer? Explain.</p>
<p>Is the absolute value inequality or equation always, sometimes, or never true? Explain.</p><p><code class='latex inline'>\displaystyle |x|=x </code></p>
<p>Evaluate.</p><p> <code class='latex inline'>\displaystyle |-22| </code></p>
<p>Solve each absolute value inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle |x-3| < 5 </code></p>
<p>Determine whether each statement is always, sometimes, or never true for real numbers <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle b </code>.</p><p><code class='latex inline'>\displaystyle |a+b|=|a|+|b| </code></p>
<p>Sketch the graph of each function.</p><p><code class='latex inline'>\displaystyle f(x) = |-2x| </code></p>
<p>Solve each inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle |6 y-2|+4 < 22 </code></p>
<p>Determine whether each statement is always, sometimes, or never true for real numbers <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle b </code>.</p><p><code class='latex inline'>\displaystyle |a b|=|a| \cdot|b| </code></p>
<p><code class='latex inline'>\displaystyle \begin{array}{l}\text { Graph each pair of equations on the same coordinate grid. } \\ \text { a. } y=2|x+1| ; y=|2 x+1| & \text { b. } y=5|x-2| ; y=|5 x-2|\end{array} </code></p>
<p>Find each absolute value.</p><p><code class='latex inline'>|-33|</code></p>
<p>Solve each equation.</p><p><code class='latex inline'>\displaystyle 4|3 x+4|=4 x+8 </code></p>
<p> Solve each inequality. Graph the solutions.</p><p><code class='latex inline'>\displaystyle |-2 x+1| > 2 </code></p>
<p>Solve for the unknown variable.</p><p><code class='latex inline'>\displaystyle 4|2 y-3|-1=11 </code></p>
<p>Make a table of values for each equation. Then graph the equation.</p><p><code class='latex inline'>\displaystyle y=|x|-3 </code></p>
<img src="/qimages/44262" /><p>ERROR ANALYSIS In Exercises <code class='latex inline'>\displaystyle \mathbf{4 5} </code> and 46, describe and correct the error in graphing the function.</p><img src="/qimages/44263" />
<p>Solve each equation. Check for extraneous solutions.</p><p><code class='latex inline'>\displaystyle |2 z-3|=4 z-1 </code></p>
<p>Solve each inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle 3|5 t-1|+9 \leq 23 </code></p>
<p>Find the vertex and the axis of symmetry of the graph of each function.</p><p><code class='latex inline'>\displaystyle y=2|x+4|-3 </code></p>
<p>Solve each inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle |2 x+1| \geq-9 </code></p>
<p>Solve each equation. Check your answers.</p><p><code class='latex inline'>\displaystyle |3 x+4|=-3 </code></p>
<p>Solve each inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle |x-5| \geq 8 </code></p>
<p>Graph each absolute value equation.</p><p><code class='latex inline'>\displaystyle y=\left|\frac{3}{2} x+2\right| </code></p>
<p>Compare and Contrast How is the graph of <code class='latex inline'>\displaystyle y=x </code> different from the graph of <code class='latex inline'>\displaystyle y=|x| ? </code></p>
<p>Solve each equation.</p><p><code class='latex inline'>\displaystyle 6|2 x+5|=6 x+24 </code></p>
<p>Solve each equation. Check your answers.</p><p><code class='latex inline'>\displaystyle |x-3|=9 </code></p>
<p>Error Analysis A classmate says that the graphs of <code class='latex inline'>\displaystyle y=-3|x| </code> and <code class='latex inline'>\displaystyle y=|-3 x| </code> are identical. Graph each function and explain why your classmate is not correct.</p>
<p>Is the absolute value inequality or equation always, sometimes, or never true? Explain.</p><p><code class='latex inline'>\displaystyle |x+2|=x+2 </code></p>
<p>a. Graph the equations <code class='latex inline'>\displaystyle f(x)=-\frac{1}{2}|x-3| </code> and <code class='latex inline'>\displaystyle g(x)=\left|-\frac{1}{2}(x-3)\right| </code> on the same set of axes.</p><p>b. Writing Describe the similarities and differences in the graphs.</p>
<p>What number is a solution to both <code class='latex inline'>\displaystyle |x-3|=2 </code> and <code class='latex inline'>\displaystyle |9-x|=8 </code> ?</p>
<img src="/qimages/44259" /><p>MODELING WITH MATHEMATICS On the pool table shown, you bank the five ball off the side represented by the <code class='latex inline'>\displaystyle x </code> -axis. The path of the ball is described by the function <code class='latex inline'>\displaystyle p(x)=\frac{4}{3}\left|x-\frac{5}{4}\right| </code>.</p><p>a. At what point does the five ball bank off the side?</p><p>b. Do you make the shot? Explain your reasoning.</p>
<p>Solve each inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle |2 x+2|-5 < 15 </code></p>
<p>Graph each equation. Then describe the transformation from the parent function <code class='latex inline'>\displaystyle f(x)=|x| </code></p><p><code class='latex inline'>\displaystyle y=-\frac{3}{4}|x| </code></p>
<p>Find the domain values of each relation if the range is {1, 16, 36}.</p><p>a. <code class='latex inline'>y=x^2</code></p><p>b. <code class='latex inline'>y=|4x|-16</code></p><p>c. <code class='latex inline'>y=|4x-16|</code></p>
<p>Is the absolute value inequality or equation always, sometimes, or never true? Explain.</p><p><code class='latex inline'>\displaystyle |x|+|x|=2 x </code></p>
<p>Solve each equation. Check your answers.</p><p><code class='latex inline'>\displaystyle |4-z|-10=1 </code></p>
<p>Solve each inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle \frac{1}{4}|x-3|+2 < 1 </code></p>
<p>Solve each equation or inequality.</p><p><code class='latex inline'>\displaystyle |2 x|+4 < 7 </code></p>
<p>Graph each absolute value equation.</p><p><code class='latex inline'>\displaystyle y=2|x+2|-3 </code></p>
<p>Solve each equation. Check your answers.</p><p><code class='latex inline'>\displaystyle |x+4|+3=17 </code></p>
<p>Solve for the unknown variable.</p><p><code class='latex inline'>\displaystyle 2|d+4|=8 </code></p>
<p>Solve each absolute value inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle |3 x+4| > -4 x-3 </code></p>
<p>What is the solution of the inequality <code class='latex inline'>\displaystyle |x-3| \geq 5 </code> ? Graph the solution.</p>
<p>Graph each absolute value equation.</p><p><code class='latex inline'>\displaystyle y=\frac{1}{2}|x|+4|x-1| </code></p>
<p>Solve for the unknown variable and graph your solution set on the number line.</p><p><code class='latex inline'>\displaystyle |4 x+7| > 19 </code></p>
<p>Solve each equation. Check for extraneous solutions.</p><p><code class='latex inline'>\displaystyle |2 y-4|=12 </code></p>
<p>Given the function <code class='latex inline'>f(x) = x^3 - 2x</code>, sketch <code class='latex inline'>y = f(|x|)</code>.</p>
<p>Solve for x.</p><p><code class='latex inline'>\displaystyle a|b x-c|=d </code></p>
<p>Graph <code class='latex inline'>f(x) = 2|x + 3| -1</code>, and state the domain and range.</p>
<p>Arrange the following values in order, from least to greatest:</p><p><code class='latex inline'>\displaystyle |-3|, -|3|, |5|, |-4|, |0| </code></p>
<p>Without graphing, identify the vertex, axis of symmetry, and transformations from the parent function <code class='latex inline'>\displaystyle f(x)=|x| </code></p><p><code class='latex inline'>\displaystyle y=|x-2|-6 </code></p>
<p>Find the vertex and the axis of symmetry of the graph of each function.</p><p><code class='latex inline'>\displaystyle y=|-x-3|+9 </code></p>
<p>Solve each absolute value inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle |2 a-1| \geq 2 a+1 </code></p>
<p>Solve for x.</p><p><code class='latex inline'>\displaystyle |c x-d|=a b </code></p>
<p><code class='latex inline'>\displaystyle |3 x|=18 </code></p>
<p>How many whole-number solutions does the</p><p> inequality <code class='latex inline'> |x-5| \leq 8 </code> have?</p>
<p>If <code class='latex inline'>\displaystyle p </code> is an integer, what is the least possible value of <code class='latex inline'>\displaystyle p </code> in the following inequality? <code class='latex inline'>\displaystyle |3 p-5| \leq 7 </code></p>
<p>OPEN ENDED Provide one example of an equation involving the Distributive Property that has no solution and another example that has infinitely many solutions.</p>
<p>Make a table of values for each equation. Then graph the equation.</p><p><code class='latex inline'>\displaystyle y=|x-5|+4 </code></p>
<p>What is the positive solution of <code class='latex inline'>\displaystyle |3 x+8|=19 ? </code></p>
<p>Solve each inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle |3 z|-4 > 8 </code></p>
<p>Graph each absolute value equation.</p><p><code class='latex inline'>\displaystyle y=6-|3 x+1| </code></p>
<p>Make a table of values for each equation. Then graph the equation.</p><p><code class='latex inline'>\displaystyle y=|x-1|+3 </code></p>
<p>Without graphing, identify the vertex, axis of symmetry, and transformations from the parent function <code class='latex inline'>\displaystyle f(x)=|x| </code></p><p><code class='latex inline'>\displaystyle y=4-|x+2| </code></p>
<p>What transformations were applied to <code class='latex inline'>\displaystyle y=|x| </code> to obtain the equation <code class='latex inline'>\displaystyle y=\left|\frac{1}{3}(x-2)\right| ? </code> a) horizontal compression by a factor of <code class='latex inline'>\displaystyle \frac{1}{3} </code>, horizontal translation 2 units to the left b) horizontal stretch by a factor of 3 , horizontal translation 2 units to the right c) horizontal translation 2 units to the right, horizontal stretch by a factor of 3</p><p>d) horizontal translation 2 units to the right, horizontal compression by a factor of <code class='latex inline'>\displaystyle \frac{1}{3} </code></p>
<p>Without graphing, identify the vertex, axis of symmetry, and transformations from the parent function <code class='latex inline'>\displaystyle f(x)=|x| </code></p><p><code class='latex inline'>\displaystyle y=\frac{3}{2}|x-6| </code></p>
<p>Make a table of values for each equation. Then graph the equation.</p><p><code class='latex inline'>\displaystyle y=|x|+1 </code></p>
<p>Graph each solution.</p><p><code class='latex inline'>\displaystyle |x| \geq 5 </code> and <code class='latex inline'>\displaystyle |x| \leq 6 </code></p>
<p>Write an absolute value function whose graph forms a square with the given graph.</p><img src="/qimages/44266" />
<p>Make a table of values for each equation. Then graph the equation.</p><p><code class='latex inline'>\displaystyle y=|x+4| </code></p>
<p> Solve each inequality. Graph the solutions.</p><p><code class='latex inline'>\displaystyle \left|\frac{x+5}{3}\right|-3 > 6 </code></p>
<p>Solve each absolute value inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle -\frac{1}{3}|f+3|+2 \geq-5 </code></p>
<p>Graph each absolute value equation.</p><p><code class='latex inline'>\displaystyle y=\left|-\frac{1}{4} x-1\right| </code></p>
<p>Sketch the graph of each function.</p><p><code class='latex inline'>\displaystyle f(x) = |x| - 2 </code></p>
<p>Graph <code class='latex inline'>\displaystyle y=4|x-3|+1 </code>. List the vertex and the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-intercepts, if any.</p>
<p>Simplify each expression. <code class='latex inline'>\displaystyle |-7+2|-4 </code></p>
<p>The graph at the right shows the translation of the graph of the parent function <code class='latex inline'>\displaystyle y=|x| </code> down 2 units and 3 units to the right. What is the area of the shaded triangle in square units?</p><img src="/qimages/94873" />
<p>For each of the following equations, state the parent function and the transformations that were applied. Graph the transformed function.</p><p><code class='latex inline'>\displaystyle f(x) =|x + 1| </code></p>
<p>Solve each equation.</p><p><code class='latex inline'>\displaystyle 5|6-5 x|=15 x-35 </code></p>
<p>Find each absolute value.</p><p><code class='latex inline'>|2.5|</code></p>
<p>Simplify each expression. <code class='latex inline'>\displaystyle 5+|4-6| </code></p>
<p>Find each absolute value.</p><p><code class='latex inline'>|77|</code></p>
<p>Solve for x.</p><p><code class='latex inline'>\displaystyle |a x|-b=c </code></p>
<p>Solve each absolute value inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle 3|d-4| \leq 13-d </code></p>
<p>Solve each equation. Check your answers.</p><p><code class='latex inline'>\displaystyle |x-2|=4 x+8 </code></p>
<p>Is the absolute value inequality or equation always, sometimes, or never true? Explain.</p><p><code class='latex inline'>\displaystyle (|x|)^{2} < x^{2} </code></p>
<p>Graph each absolute value equation.</p><p><code class='latex inline'>\displaystyle y=-|x-3| </code></p>
<p> Solve each inequality. Graph the solutions.</p><p><code class='latex inline'>\displaystyle \frac{1}{11}|2 x-4|+10 \leq 11 </code></p>
<p>Match the absolute value equation with its graph without using the equation. </p><img src="/qimages/42690" /><p><code class='latex inline'>\displaystyle |x+2|=4 </code></p>
<p>Which number is a solution of <code class='latex inline'>\displaystyle |9-x|=9+x </code> ? (F) <code class='latex inline'>\displaystyle -3 </code></p>
<p>Describe the possible values for <code class='latex inline'>x</code> and <code class='latex inline'>y</code> when <code class='latex inline'>|x-y|>0</code>. What does it mean when <code class='latex inline'>|x-y|=0</code>? Can <code class='latex inline'>|x-y|<0</code>? Explain your reasoning. </p>
<p>Graph each equation. Then describe the transformation from the parent function <code class='latex inline'>\displaystyle f(x)=|x| </code></p><p><code class='latex inline'>\displaystyle y=\frac{3}{2}|x| </code></p>
<p>Solve each equation.</p><p><code class='latex inline'>\displaystyle \frac{1}{2}|3 c+5|=6 c+4 </code></p>
<p>Sketch the graph of each function.</p><p><code class='latex inline'>\displaystyle f(x) = |0.5x| </code></p>
<p>Solve for the unknown variable and graph your solution set on the number line.</p><p><code class='latex inline'>\displaystyle |-3 n|-2=4 </code></p>
<p>Determine whether each statement is always, sometimes, or never true for real numbers <code class='latex inline'>\displaystyle a </code> and <code class='latex inline'>\displaystyle b </code>.</p><p><code class='latex inline'>\displaystyle \left|\frac{a}{b}\right|=\frac{|a|}{|b|}</code>, <code class='latex inline'>b \neq 0 </code></p>
<p>Graph each equation. Then describe the transformation from the parent function <code class='latex inline'>\displaystyle f(x)=|x| </code></p><p><code class='latex inline'>\displaystyle y=-\frac{1}{2}|x| </code></p>
<p>Solve each absolute value inequality. Graph the solution.</p><p><code class='latex inline'>\displaystyle |3 x+1|+1 > 12 </code></p>
<p> Solve each inequality. Graph the solutions.</p><p><code class='latex inline'>\displaystyle 2|4 t-1|+6 > 20 </code></p>
<p>Solve each equation. Check your answers.</p><p><code class='latex inline'>\displaystyle |-6 x|=24 </code></p>
<p>Graph each solution.</p><p><code class='latex inline'>\displaystyle |x-5| \leq x </code></p>
<p>Solve each equation.</p><p><code class='latex inline'>\displaystyle \frac{2}{3}|3 x-6|=4(x-2) </code></p>
<p>Sketch the graph of each function.</p><p><code class='latex inline'>\displaystyle f(x) = |x| + 3 </code></p>
<p>Think About a Plan Graph <code class='latex inline'>\displaystyle y=-2|x+3|+4 </code>. List the <code class='latex inline'>\displaystyle x </code> - and <code class='latex inline'>\displaystyle y </code>-intercepts, if any.</p> <ul> <li><p>What is the vertex?</p></li> <li><p>What does <code class='latex inline'>\displaystyle y </code> equal at the <code class='latex inline'>\displaystyle x </code>-intercept(s)? What does <code class='latex inline'>\displaystyle x </code> equal at the <code class='latex inline'>\displaystyle y </code>-intercept(s)?</p></li> </ul>
<p>Is the absolute value inequality or equation always, sometimes, or never true? Explain.</p><p><code class='latex inline'>\displaystyle |x|=-6 </code></p>
<img src="/qimages/44262" /><p>ERROR ANALYSIS In Exercises <code class='latex inline'>\displaystyle \mathbf{4 5} </code> and 46, describe and correct the error in graphing the function.</p><img src="/qimages/44264" /><p><code class='latex inline'>\displaystyle y=-3|x| </code></p>
<p>Is <code class='latex inline'>\displaystyle |y|=x </code> a function? Explain.</p>
<p>Graph each solution.</p><p><code class='latex inline'>\displaystyle |x| \geq 6 </code> or <code class='latex inline'>\displaystyle |x| < 5 </code></p>
<p>Solve each equation. Check for extraneous solutions.</p><p><code class='latex inline'>\displaystyle |2 x+5|=3 x+4 </code></p>
<p>Write an absolute value function whose graph forms a square with the given graph.</p><img src="/qimages/44267" />
<p> Solve each inequality. Graph the solutions.</p><p><code class='latex inline'>\displaystyle |2 x+3|-6 \geq 7 </code></p>
<p>Solve each equation or inequality.</p><p><code class='latex inline'>\displaystyle |3 x+9|=11 </code></p>
<p>Determine if each function is a vertical stretch or vertical compression of the parent function <code class='latex inline'>\displaystyle y=|x| </code>.</p><p><code class='latex inline'>\displaystyle y=\frac{3}{2}|x| </code></p>
<p>The graph at the right models the distance between a roadside stand and a car traveling at a constant speed. The <code class='latex inline'>\displaystyle x </code>-axis represents time and the <code class='latex inline'>\displaystyle y </code>-axis represents distance. Which equation best represents the relation shown in the graph? <code class='latex inline'>\displaystyle \begin{array}{ll}\text { A } y=|60 x| & \text { C } y=|x|+60 \\ \text { B } y=|40 x| & \text { (D } y=|x|+40\end{array} </code></p><img src="/qimages/89619" />
How did you do?
I failed
I think I failed
I think I got it
I got it
Another question?
Found an error or missing video? We'll update it within the hour! 👉
Report it
Save videos to My Cheatsheet for later, for easy studying.