Exponential Inequalities

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Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Algebra II EOC · 2013 · Item 30 · 1 point

Worked example

Use a table and graph reasoning to solve $f(x)>24$ for $f(x)=3(2^x)$.

1. Compute nearby outputs: $f(2)=12$, $f(3)=24$, and $f(4)=48$.
2. The curve intersects the target line $y=24$ at $x=3$.
3. The base 2 makes the function increasing, so it stays above the target to the right.
4. The inequality is strict, so exclude the crossing input.

$x>3$

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

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TEKSForge practice · unscored

1. 1. Solve $10(3^x)\le90$.

   Show answer and reasoning

   $x\le2$

   The increasing function equals 90 at $x=2$. Outputs at or below 90 occur at or left of that input.

2. 2. A quantity is $Q(d)=64\left(\frac12\right)^d$, where d counts completed days beginning at 0. On which days is $Q(d)<8$?

   Show answer and reasoning

   Whole-number days $d\ge4$.

   $Q(3)=8$, and the function decreases. The continuous solution is $d>3$; completed days begin at 4.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 30 · 1 point

The graph of the exponential function $f$ is shown on the grid. For what values of $x$ is $f(x) > 16$?

1. $x > -2$
2. $x > 0$
3. $-2 < x < 2$
4. $\infty < x < -2$

Show STAAR answer and explanation

$x > -2$

The increasing curve passes through $(-2,16)$. Because f increases everywhere, its values are greater than 16 for every x to the right of −2.

$x > 0$: At $x=0$, $f(0)$ is about 32; the values are already above 16 for $-2<x<0$.

$-2 < x < 2$: The curve keeps rising after $x=2$, so there is no upper bound on x.

$\infty < x < -2$: To the left of $x=-2$, the values are less than 16.

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